From fd34fe197dbf07961296b997b19bb6d855c5d897 Mon Sep 17 00:00:00 2001 From: 0000OOOO0000 <63518686+0000OOOO0000@users.noreply.github.com> Date: Sat, 23 Oct 2021 18:36:35 +0300 Subject: [PATCH] Add files via upload --- ...”“α”•βœ€κ–΄α‘α‘•α—©β €β—―β €β €β €β €.GHX | 20039 ++++++++++++++++ 1 file changed, 20039 insertions(+) create mode 100644 β—―α—©IᗝIβš­β—―βšͺβ—―βš­IᗝIα—©β—―β΅™β—―α—©IᗝIβš­β—―βšͺβ—―βš­IᗝIα—©β—―/β—―βœ€α΄₯α—©β—―β΅™β—―α—©α΄₯βœ€β—―/β—―α—±α—΄α΄₯α—©α—―βœ€β€β“„α”“α”•β—―β΅™β—―α”“α”•β“„β€βœ€α—―α—©α΄₯α—±α—΄β—―/β—―α—β΅ˆβ—―β΅™β—―β΅ˆα—β—―/β—―α”“α”•β“„α΄₯α—±α—΄α‘α‘•β“„Π˜Nκ–΄μ˜·α΄₯β—―βšͺβ—―α΄₯μ˜·κ–΄Π˜NⓄᑐᑕᗱᗴα΄₯β“„α”“α”•β—―β΅™β—―α”“α”•β“„α΄₯α—±α—΄α‘α‘•β“„Π˜Nκ–΄μ˜·α΄₯β—―βšͺβ—―α΄₯μ˜·κ–΄Π˜NⓄᑐᑕᗱᗴα΄₯β“„α”“α”•β—―/β—―α΄₯α—±α—΄ί¦β“„μ˜·α”“α”•α—©α΄₯ᕀᕦ◯βšͺ◯ᕀᕦα΄₯α—©α”“α”•μ˜·β“„ί¦α—±α—΄α΄₯β—―β΅™β—―α΄₯α—±α—΄ί¦β“„μ˜·α”“α”•α—©α΄₯ᕀᕦ◯βšͺ◯ᕀᕦα΄₯α—©α”“α”•μ˜·β“„ί¦α—±α—΄α΄₯β—―/XHG.β €β €β €β €β—―β €α—©α‘α‘•κ–΄βœ€α”“α”•α—©α™α—±α—΄β €β—―β €β €β €β €β΅™β €β €β €β €β—―β €α—±α—΄α™α—©α”“α”•βœ€κ–΄α‘α‘•α—©β €β—―β €β €β €β €.GHX diff --git a/β—―α—©IᗝIβš­β—―βšͺβ—―βš­IᗝIα—©β—―β΅™β—―α—©IᗝIβš­β—―βšͺβ—―βš­IᗝIα—©β—―/β—―βœ€α΄₯α—©β—―β΅™β—―α—©α΄₯βœ€β—―/β—―α—±α—΄α΄₯α—©α—―βœ€β€β“„α”“α”•β—―β΅™β—―α”“α”•β“„β€βœ€α—―α—©α΄₯α—±α—΄β—―/β—―α—β΅ˆβ—―β΅™β—―β΅ˆα—β—―/β—―α”“α”•β“„α΄₯α—±α—΄α‘α‘•β“„Π˜Nκ–΄μ˜·α΄₯β—―βšͺβ—―α΄₯μ˜·κ–΄Π˜NⓄᑐᑕᗱᗴα΄₯β“„α”“α”•β—―β΅™β—―α”“α”•β“„α΄₯α—±α—΄α‘α‘•β“„Π˜Nκ–΄μ˜·α΄₯β—―βšͺβ—―α΄₯μ˜·κ–΄Π˜NⓄᑐᑕᗱᗴα΄₯β“„α”“α”•β—―/β—―α΄₯α—±α—΄ί¦β“„μ˜·α”“α”•α—©α΄₯ᕀᕦ◯βšͺ◯ᕀᕦα΄₯α—©α”“α”•μ˜·β“„ί¦α—±α—΄α΄₯β—―β΅™β—―α΄₯α—±α—΄ί¦β“„μ˜·α”“α”•α—©α΄₯ᕀᕦ◯βšͺ◯ᕀᕦα΄₯α—©α”“α”•μ˜·β“„ί¦α—±α—΄α΄₯β—―/XHG.β €β €β €β €β—―β €α—©α‘α‘•κ–΄βœ€α”“α”•α—©α™α—±α—΄β €β—―β €β €β €β €β΅™β €β €β €β €β—―β €α—±α—΄α™α—©α”“α”•βœ€κ–΄α‘α‘•α—©β €β—―β €β €β €β €.GHX b/β—―α—©IᗝIβš­β—―βšͺβ—―βš­IᗝIα—©β—―β΅™β—―α—©IᗝIβš­β—―βšͺβ—―βš­IᗝIα—©β—―/β—―βœ€α΄₯α—©β—―β΅™β—―α—©α΄₯βœ€β—―/β—―α—±α—΄α΄₯α—©α—―βœ€β€β“„α”“α”•β—―β΅™β—―α”“α”•β“„β€βœ€α—―α—©α΄₯α—±α—΄β—―/β—―α—β΅ˆβ—―β΅™β—―β΅ˆα—β—―/β—―α”“α”•β“„α΄₯α—±α—΄α‘α‘•β“„Π˜Nκ–΄μ˜·α΄₯β—―βšͺβ—―α΄₯μ˜·κ–΄Π˜NⓄᑐᑕᗱᗴα΄₯β“„α”“α”•β—―β΅™β—―α”“α”•β“„α΄₯α—±α—΄α‘α‘•β“„Π˜Nκ–΄μ˜·α΄₯β—―βšͺβ—―α΄₯μ˜·κ–΄Π˜NⓄᑐᑕᗱᗴα΄₯β“„α”“α”•β—―/β—―α΄₯α—±α—΄ί¦β“„μ˜·α”“α”•α—©α΄₯ᕀᕦ◯βšͺ◯ᕀᕦα΄₯α—©α”“α”•μ˜·β“„ί¦α—±α—΄α΄₯β—―β΅™β—―α΄₯α—±α—΄ί¦β“„μ˜·α”“α”•α—©α΄₯ᕀᕦ◯βšͺ◯ᕀᕦα΄₯α—©α”“α”•μ˜·β“„ί¦α—±α—΄α΄₯β—―/XHG.β €β €β €β €β—―β €α—©α‘α‘•κ–΄βœ€α”“α”•α—©α™α—±α—΄β €β—―β €β €β €β €β΅™β €β €β €β €β—―β €α—±α—΄α™α—©α”“α”•βœ€κ–΄α‘α‘•α—©β €β—―β €β €β €β €.GHX new file mode 100644 index 00000000..c93e386e --- /dev/null +++ b/β—―α—©IᗝIβš­β—―βšͺβ—―βš­IᗝIα—©β—―β΅™β—―α—©IᗝIβš­β—―βšͺβ—―βš­IᗝIα—©β—―/β—―βœ€α΄₯α—©β—―β΅™β—―α—©α΄₯βœ€β—―/β—―α—±α—΄α΄₯α—©α—―βœ€β€β“„α”“α”•β—―β΅™β—―α”“α”•β“„β€βœ€α—―α—©α΄₯α—±α—΄β—―/β—―α—β΅ˆβ—―β΅™β—―β΅ˆα—β—―/β—―α”“α”•β“„α΄₯α—±α—΄α‘α‘•β“„Π˜Nκ–΄μ˜·α΄₯β—―βšͺβ—―α΄₯μ˜·κ–΄Π˜NⓄᑐᑕᗱᗴα΄₯β“„α”“α”•β—―β΅™β—―α”“α”•β“„α΄₯α—±α—΄α‘α‘•β“„Π˜Nκ–΄μ˜·α΄₯β—―βšͺβ—―α΄₯μ˜·κ–΄Π˜NⓄᑐᑕᗱᗴα΄₯β“„α”“α”•β—―/β—―α΄₯α—±α—΄ί¦β“„μ˜·α”“α”•α—©α΄₯ᕀᕦ◯βšͺ◯ᕀᕦα΄₯α—©α”“α”•μ˜·β“„ί¦α—±α—΄α΄₯β—―β΅™β—―α΄₯α—±α—΄ί¦β“„μ˜·α”“α”•α—©α΄₯ᕀᕦ◯βšͺ◯ᕀᕦα΄₯α—©α”“α”•μ˜·β“„ί¦α—±α—΄α΄₯β—―/XHG.β €β €β €β €β—―β €α—©α‘α‘•κ–΄βœ€α”“α”•α—©α™α—±α—΄β €β—―β €β €β €β €β΅™β €β €β €β €β—―β €α—±α—΄α™α—©α”“α”•βœ€κ–΄α‘α‘•α—©β €β—―β €β €β €β €.GHX @@ -0,0 +1,20039 @@ +ο»Ώ + + + + + + + 0 + 2 + 2 + + + + + + + 1 + 0 + 7 + + + + + + 1c961c8b-9745-4a92-a43d-080de1ead765 + Shaded + 1 + + 100;150;0;0 + + + 100;0;150;0 + + + + + + 635273898765795129 + + elastica_curve_examples - Copy.ghx + + + + + 0 + + + + + + -102 + 40 + + 1 + + + + + 0 + + + + + + + 0 + + + + + 149 + + + + + c552a431-af5b-46a9-a8a4-0fcbc27ef596 + Group + + + + + 1 + + 150;170;135;255 + + A group of Grasshopper objects + d013dc08-a8cd-4383-aa4a-7e9f0b202f67 + 19d4e5e6-a3fb-4e4d-b426-93c0b41f974c + ce2f14ec-483c-4899-a8cb-784a62168957 + b2a67d0f-c66e-46a9-8efd-f7442d233d5d + 32bb1a9f-9575-4b8c-8a60-a65a7b9dd15f + 98102773-859e-4cf3-83a5-41f68379af66 + d68f5884-1ed1-4bd5-ab64-b7040370d59b + 8cd6ad76-7f71-4948-8c5e-9a3e2549985f + d53a1087-053a-44d5-b485-68a8b5d09ce4 + cc8dfb80-5022-4b13-83c9-a787888900e8 + 072c5f2f-5efd-4587-8eb9-f4eacb6f59a9 + 25d0b3b4-fc42-4433-a4bf-e70bfa828143 + 5137ef09-783f-4981-a9ec-aa4f2fc8e019 + 225afdb2-480f-435b-a1cb-84170b3afd2f + 20228f31-e357-4d65-8747-46e12348391c + 7c2a1ac2-4916-4aa3-9b0a-566a67f36e60 + 95f9fd7f-37dc-4bd8-8105-7301ef052bdd + 17 + 07a70634-4e1a-4226-b5d3-17b0a4e0f460 + Group + + + + + + + + + + + 079bd9bd-54a0-41d4-98af-db999015f63d + VB Script + + + + + Private Function IsSet(ByVal param As String) As Boolean ' Check if an input parameter has data + Dim i As Integer = Component.Params.IndexOfInputParam(param) + If i > -1 Then + Return Component.Params.Input.ElementAt(i).DataType > 1 ' input parameter DataType of 1 means it's not receiving input (internal or external) + Else + Msg("error", "Input parameter '" & param & "' not found") + Return False + End If + End Function + + Private Sub Msg(ByVal type As String, ByVal msg As String) ' Output an error, warning, or informational message + Select Case type + Case "error" + Component.AddRuntimeMessage(GH_RuntimeMessageLevel.Error, msg) + Print("Error: " & msg) + Case "warning" + Component.AddRuntimeMessage(GH_RuntimeMessageLevel.Warning, msg) + Print("Warning: " & msg) + Case "info" + Component.AddRuntimeMessage(GH_RuntimeMessageLevel.Remark, msg) + Print(msg) + End Select + End Sub + + ' Solve for the m parameter from length and width (reference {1} equation (34), except b = width and K(k) and E(k) should be K(m) and E(m)) + Private Function SolveMFromLenWid(ByVal L As Double, ByVal w As Double) As Double + If w = 0 Then + Return Defined.M_ZERO_W ' for the boundry condition width = 0, bypass the function and return the known m value + End If + + Dim n As Integer = 1 ' Iteration counter (quit if >MAXIT) + Dim lower As Double = 0 ' m must be within this range + Dim upper As Double = 1 + Dim m As Double + Dim cwl As Double + + Do While (upper - lower) > Defined.MAXERR AndAlso (n) < Defined.MAXIT ' Repeat until range narrow enough or MAXIT + m = (upper + lower) / 2 + cwl = 2 * EllipticE(m) / EllipticK(m) - 1 ' calculate w/L with the test value of m + If cwl < w / L Then ' compares the calculated w/L with the actual w/L then narrows the range of possible m + upper = m + Else + lower = m + End If + n += 1 + Loop + Return m + End Function + + ' Solve for the m parameter from length and height (reference {1} equation (33), except K(k) should be K(m) and k = sqrt(m)) + ' Note that it's actually possible to find 2 valid values for m (hence 2 width values) at certain height values + Private Function SolveMFromLenHt(ByVal L As Double, ByVal h As Double) As List(Of Double) + Dim n As Integer = 1 ' Iteration counter (quit if >MAXIT) + Dim lower As Double = 0 ' m must be within this range + Dim upper As Double = 1 + Dim twoWidths As Boolean = h / L >= Defined.DOUBLE_W_HL_RATIO And h / L < Defined.MAX_HL_RATIO ' check to see if h/L is within the range where 2 solutions for the width are possible + Dim m As Double + Dim mult_m As New List(Of Double) + Dim chl As Double + + If twoWidths Then + ' find the first of two possible solutions for m with the following limits: + lower = Defined.M_DOUBLE_W ' see constants at bottom of script + upper = Defined.M_MAXHEIGHT ' see constants at bottom of script + Do While (upper - lower) > Defined.MAXERR AndAlso (n) < Defined.MAXIT ' Repeat until range narrow enough or MAXIT + m = (upper + lower) / 2 + chl = Math.Sqrt(m) / EllipticK(m) ' calculate h/L with the test value of m + If chl > h / L Then ' compares the calculated h/L with the actual h/L then narrows the range of possible m + upper = m + Else + lower = m + End If + n += 1 + Loop + mult_m.Add(m) + + ' then find the second of two possible solutions for m with the following limits: + lower = Defined.M_MAXHEIGHT ' see constants at bottom of script + upper = 1 + Do While (upper - lower) > Defined.MAXERR AndAlso (n) < Defined.MAXIT ' Repeat until range narrow enough or MAXIT + m = (upper + lower) / 2 + chl = Math.Sqrt(m) / EllipticK(m) ' calculate h/L with the test value of m + If chl < h / L Then ' compares the calculated h/L with the actual h/L then narrows the range of possible m + upper = m + Else + lower = m + End If + n += 1 + Loop + + If m <= Defined.M_MAX Then ' return this m parameter only if it falls within the maximum useful value (above which the curve breaks down) + mult_m.Add(m) + End If + + Else + ' find the one possible solution for the m parameter + upper = Defined.M_DOUBLE_W ' limit the upper end of the search to the maximum value of m for which only one solution exists + Do While (upper - lower) > Defined.MAXERR AndAlso (n) < Defined.MAXIT ' Repeat until range narrow enough or MAXIT + m = (upper + lower) / 2 + chl = Math.Sqrt(m) / EllipticK(m) ' calculate h/L with the test value of m + If chl > h / L Then ' compares the calculated h/L with the actual h/L then narrows the range of possible m + upper = m + Else + lower = m + End If + n += 1 + Loop + mult_m.Add(m) + End If + + Return mult_m + End Function + + ' Solve for the m parameter from width and height (derived from reference {1} equations (33) and (34) with same notes as above) + Private Function SolveMFromWidHt(ByVal w As Double, ByVal h As Double) As Double + Dim n As Integer = 1 ' Iteration counter (quit if >MAXIT) + Dim lower As Double = 0 ' m must be within this range + Dim upper As Double = 1 + Dim m As Double + Dim cwh As Double + + Do While (upper - lower) > Defined.MAXERR AndAlso (n) < Defined.MAXIT ' Repeat until range narrow enough or MAXIT + m = (upper + lower) / 2 + cwh = (2 * EllipticE(m) - EllipticK(m)) / Math.Sqrt(m) ' calculate w/h with the test value of m + If cwh < w / h Then ' compares the calculated w/h with the actual w/h then narrows the range of possible m + upper = m + Else + lower = m + End If + n += 1 + Loop + + Return m + End Function + + ' Calculate length based on height and an m parameter, derived from reference {1} equation (33), except K(k) should be K(m) and k = sqrt(m) + Private Function Cal_L(ByVal h As Double, ByVal m As Double) As Double + Return h * EllipticK(m) / Math.Sqrt(m) + End Function + + ' Calculate width based on length and an m parameter, derived from reference {1} equation (34), except b = width and K(k) and E(k) should be K(m) and E(m) + Private Function Cal_W(ByVal L As Double, ByVal m As Double) As Double + Return L * (2 * EllipticE(m) / EllipticK(m) - 1) + End Function + + ' Calculate height based on length and an m parameter, from reference {1} equation (33), except K(k) should be K(m) and k = sqrt(m) + Private Function Cal_H(ByVal L As Double, ByVal m As Double) As Double + Return L * Math.Sqrt(m) / EllipticK(m) + End Function + + ' Calculate the unique m parameter based on a start tangent angle, from reference {2}, just above equation (9a), that states k = Sin(angle / 2 + Pi / 4), + ' but as m = k^2 and due to this script's need for an angle rotated 90Β° versus the one in reference {1}, the following formula is the result + ' New note: verified by reference {4}, pg. 78 at the bottom + Private Function Cal_M(ByVal a As Double) As Double + Return (1 - Math.Cos(a)) / 2 ' equal to Sin^2(a/2) too + End Function + + ' Calculate start tangent angle based on an m parameter, derived from above formula + Private Function Cal_A(ByVal m As Double) As Double + Return Math.Acos(1 - 2 * m) + End Function + + ' This is the heart of this script, taking the found (or specified) length, width, and angle values along with the found m parameter to create + ' a list of points that approximate the shape or form of the elastica. It works by finding the x and y coordinates (which are reversed versus + ' the original equations (12a) and (12b) from reference {2} due to the 90Β° difference in orientation) based on the tangent angle along the curve. + ' See reference {2} for more details on how they derived it. Note that to simplify things, the algorithm only calculates the points for half of the + ' curve, then mirrors those points along the y-axis. + Private Function FindBendForm(ByVal L As Double, ByVal w As Double, ByVal m As Double, ByVal ang As Double, ByVal refPln As Plane) As List(Of Point3d) + L = L / 2 ' because the below algorithm is based on the formulas in reference {2} for only half of the curve + w = w / 2 ' same + + If ang = 0 Then ' if angle (and height) = 0, then simply return the start and end points of the straight line + Dim out As New List(Of Point3d) + out.Add(refPln.PointAt(w, 0, 0)) + out.Add(refPln.PointAt(-w, 0, 0)) + Return out + End If + + Dim x As Double + Dim y As Double + Dim halfCurvePts As New List(Of Point3d) + Dim fullCurvePts As New List(Of Point3d) + Dim translatedPts As New List(Of Point3d) + + ang -= Math.PI / 2 ' a hack to allow this algorithm to work, since the original curve in paper {2} was rotated 90Β° + Dim angB As Double = ang + (-Math.PI / 2 - ang) / Defined.CURVEDIVS ' angB is the 'lowercase theta' which should be in formula {2}(12b) as the interval + ' start [a typo...see equation(3)]. It's necessary to start angB at ang + [interval] instead of just ang due to integration failing at angB = ang + halfCurvePts.Add(New Point3d(w, 0, 0)) ' start with this known initial point, as integration will fail when angB = ang + + ' each point {x, y} is calculated from the tangent angle, angB, that occurs at each point (which is why this iterates from ~ang to -pi/2, the known end condition) + Do While Math.Round(angB, Defined.ROUNDTO) >= Math.Round(-Math.PI / 2, Defined.ROUNDTO) + y = (Math.Sqrt(2) * Math.Sqrt(Math.Sin(ang) - Math.Sin(angB)) * (w + L)) / (2 * EllipticE(m)) ' note that x and y are swapped vs. (12a) and (12b) + x = (L / (Math.Sqrt(2) * EllipticK(m))) * Simpson(angB, -Math.PI / 2, 500, ang) ' calculate the Simpson approximation of the integral (function f below) + ' over the interval angB ('lowercase theta') to -pi/2. side note: is 500 too few iterations for the Simson algorithm? + + If Math.Round(x, Defined.ROUNDTO) = 0 Then x = 0 + halfCurvePts.Add(New Point3d(x, y, 0)) + + angB += (-Math.PI / 2 - ang) / Defined.CURVEDIVS ' onto the next tangent angle + Loop + + ' After finding the x and y values for half of the curve, add the {-x, y} values for the rest of the curve + For Each point As Point3d In halfCurvePts + If Math.Round(point.X, Defined.ROUNDTO) = 0 Then + If Math.Round(point.Y, Defined.ROUNDTO) = 0 Then + fullCurvePts.Add(New Point3d(0, 0, 0)) ' special case when width = 0: when x = 0, only duplicate the point when y = 0 too + End If + Else + fullCurvePts.Add(New Point3d(-point.X, point.Y, 0)) + End If + Next + halfCurvePts.Reverse + fullCurvePts.AddRange(halfCurvePts) + + For Each p As Point3d In fullCurvePts + translatedPts.Add(refPln.PointAt(p.X, p.Y, p.Z)) ' translate the points from the reference plane to the world plane + Next + + Return translatedPts + End Function + + ' Interpolates the points from FindBendForm to create the Elastica curve. Uses start & end tangents for greater accuracy. + Private Function MakeCurve(ByVal pts As List(Of Point3d), ByVal ang As Double, ByVal refPln As Plane) As Curve + If ang <> 0 Then + Dim ts, te As New Vector3d(refPln.XAxis) + ts.Rotate(ang, refPln.ZAxis) + te.Rotate(-ang, refPln.ZAxis) + Return Curve.CreateInterpolatedCurve(pts, 3, CurveKnotStyle.Chord, ts, te) ' 3rd degree curve with 'Chord' Knot Style + Else + Return Curve.CreateInterpolatedCurve(pts, 3) ' if angle (and height) = 0, then simply interpolate the straight line (no start/end tangents) + End If + End Function + + ' Implements the Simpson approximation for an integral of function f below + Public Function Simpson(a As Double, b As Double, n As Integer, theta As Double) As Double 'n should be an even number + Dim j As Integer, s1 As Double, s2 As Double, h As Double + h = (b - a) / n + s1 = 0 + s2 = 0 + For j = 1 To n - 1 Step 2 + s1 = s1 + fn(a + j * h, theta) + Next j + For j = 2 To n - 2 Step 2 + s2 = s2 + fn(a + j * h, theta) + Next j + Simpson = h / 3 * (fn(a, theta) + 4 * s1 + 2 * s2 + fn(b, theta)) + End Function + + ' Specific calculation for the above integration + Public Function fn(x As Double, theta As Double) As Double + fn = Math.Sin(x) / (Math.Sqrt(Math.Sin(theta) - Math.Sin(x))) ' from reference {2} formula (12b) + End Function + + + ' Return the Complete Elliptic integral of the 1st kind + ' Abramowitz and Stegun p.591, formula 17.3.11 + ' Code from http://www.codeproject.com/Articles/566614/Elliptic-integrals + Public Function EllipticK(ByVal m As Double) As Double + Dim sum, term, above, below As Double + sum = 1 + term = 1 + above = 1 + below = 2 + + For i As Integer = 1 To 100 + term *= above / below + sum += Math.Pow(m, i) * Math.Pow(term, 2) + above += 2 + below += 2 + Next + sum *= 0.5 * Math.PI + Return sum + End Function + + + ' Return the Complete Elliptic integral of the 2nd kind + ' Abramowitz and Stegun p.591, formula 17.3.12 + ' Code from http://www.codeproject.com/Articles/566614/Elliptic-integrals + Public Function EllipticE(ByVal m As Double) As Double + Dim sum, term, above, below As Double + sum = 1 + term = 1 + above = 1 + below = 2 + + For i As Integer = 1 To 100 + term *= above / below + sum -= Math.Pow(m, i) * Math.Pow(term, 2) / above + above += 2 + below += 2 + Next + sum *= 0.5 * Math.PI + Return sum + End Function + + Friend Partial NotInheritable Class Defined + Private Sub New() + End Sub + + ' Note: most of these values for m and h/L ratio were found with Wolfram Alpha and either specific intercepts (x=0) or local minima/maxima. They should be constant. + Public Const M_SKETCHY As Double = 0.95 ' value of the m parameter where the curvature near the ends of the curve gets wonky + Public Const M_MAX As Double = 0.993 ' maximum useful value of the m parameter, above which this algorithm for the form of the curve breaks down + Public Const M_ZERO_W As Double = 0.826114765984970336 ' value of the m parameter when width = 0 + Public Const M_MAXHEIGHT As Double = 0.701327460663101223 ' value of the m parameter at maximum possible height of the bent rod/wire + Public Const M_DOUBLE_W As Double = 0.180254422335013983 ' minimum value of the m parameter when two width values are possible for a given height and length + Public Const DOUBLE_W_HL_RATIO As Double = 0.257342117984635757 ' value of the height/length ratio above which there are two possible width values + Public Const MAX_HL_RATIO As Double = 0.403140189705650243 ' maximum possible value of the height/length ratio + + Public Const MAXERR As Double = 0.0000000001 ' error tolerance + Public Const MAXIT As Integer = 100 ' maximum number of iterations + Public Const ROUNDTO As Integer = 10 ' number of decimal places to round off to + Public Const CURVEDIVS As Integer = 50 ' number of sample points for building the curve (or half-curve as it were) + End Class + A VB.NET scriptable component + + 98 + 86 + + true + d013dc08-a8cd-4383-aa4a-7e9f0b202f67 + VB Script + VB + true + 0 + ' ----------------------------------------------------------------- + ' Elastic Bending Script by Will McElwain + ' Created February 2014 + ' + ' DESCRIPTION: + ' This beast creates the so-called 'elastica curve', the shape a long, thin rod or wire makes when it is bent elastically (i.e. not permanently). In this case, force + ' is assumed to only be applied horizontally (which would be in line with the rod at rest) and both ends are assumed to be pinned or hinged meaning they are free + ' to rotate (as opposed to clamped, when the end tangent angle is fixed, usually horizontally). An interesting finding is that it doesn't matter what the material or + ' cross-sectional area is, as long as they're uniform along the entire length. Everything makes the same shape when bent as long as it doesn't cross the threshold + ' from elastic to plastic (permanent) deformation (I don't bother to find that limit here, but can be found if the yield stress for a material is known). + ' + ' Key to the formulas used in this script are elliptic integrals, specifically K(m), the complete elliptic integral of the first kind, and E(m), the complete elliptic + ' integral of the second kind. There was a lot of confusion over the 'm' and 'k' parameters for these functions, as some people use them interchangeably, but they are + ' not the same. m = k^2 (thus k = Sqrt(m)). I try to use the 'm' parameter exclusively to avoid this confusion. Note that there is a unique 'm' parameter for every + ' configuration/shape of the elastica curve. + ' + ' This script tries to find that unique 'm' parameter based on the inputs. The algorithm starts with a test version of m, evaluates an expression, say 2*E(m)/K(m)-1, + ' then compares the result to what it should be (in this case, a known width/length ratio). Iterate until the correct m is found. Once we have m, we can then calculate + ' all of the other unknowns, then find points that lie on that curve, then interpolate those points for the actual curve. You can also use Wolfram|Alpha as I did to + ' find the m parameter based on the equations in this script (example here: http://tiny.cc/t4tpbx for when say width=45.2 and length=67.1). + ' + ' Other notes: + ' * This script works with negative values for width, which will creat a self-intersecting curve (as it should). The curvature of the elastica starts to break down around + ' m=0.95 (~154Β°), but this script will continue to work until M_MAX, m=0.993 (~169Β°). If you wish to ignore self-intersecting curves, set ignoreSelfIntersecting to True + ' * When the only known values are length and height, it is actually possible for certain ratios of height to length to have two valid m values (thus 2 possible widths + ' and angles). This script will return them both. + ' * Only the first two valid parameters (of the required ones) will be used, meaning if all four are connected (length, width or a PtB, height, and angle), this script will + ' only use length and width (or a PtB). + ' * Depending on the magnitude of your inputs (say if they're really small, like if length < 10), you might have to increase the constant ROUNDTO at the bottom + ' + ' REFERENCES: + ' {1} "The elastic rod" by M.E. Pacheco Q. & E. Pina, http://www.scielo.org.mx/pdf/rmfe/v53n2/v53n2a8.pdf + ' {2} "An experiment in nonlinear beam theory" by A. Valiente, http://www.deepdyve.com/lp/doc/I3lwnxdfGz , also here: http://tiny.cc/Valiente_AEiNBT + ' {3} "Snap buckling, writhing and Loop formation In twisted rods" by V.G.A. GOSS, http://myweb.lsbu.ac.uk/~gossga/thesisFinal.pdf + ' {4} "Theory of Elastic Stability" by Stephen Timoshenko, http://www.scribd.com/doc/50402462/Timoshenko-Theory-of-Elastic-Stability (start on p. 76) + ' + ' INPUT: + ' PtA - First anchor point (required) + ' PtB - Second anchor point (optional, though 2 out of the 4--length, width, height, angle--need to be specified) + ' [note that PtB can be the same as PtA (meaning width would be zero)] + ' [also note that if a different width is additionally specified that's not equal to the distance between PtA and PtB, then the end point will not equal PtB anymore] + ' Pln - Plane of the bent rod/wire, which bends up in the +y direction. The line between PtA and PtB (if specified) must be parallel to the x-axis of this plane + ' + ' ** 2 of the following 4 need to be specified ** + ' Len - Length of the rod/wire, which needs to be > 0 + ' Wid - Width between the endpoints of the curve [note: if PtB is specified in addition, and distance between PtA and PtB <> width, the end point will be relocated + ' Ht - Height of the bent rod/wire (when negative, curve will bend downward, relative to the input plane, instead) + ' Ang - Inner departure angle or tangent angle (in radians) at the ends of the bent rod/wire. Set up so as width approaches length (thus height approaches zero), angle approaches zero + ' + ' * Following variables only needed for optional calculating of bending force, not for shape of curve. + ' E - Young's modulus (modulus of elasticity) in GPa (=N/m^2) (material-specific. for example, 7075 aluminum is roughly 71.7 GPa) + ' I - Second moment of area (or area moment of inertia) in m^4 (cross-section-specific. for example, a hollow rod + ' would have I = pi * (outer_diameter^4 - inner_diameter^4) / 32 + ' Note: E*I is also known as flexural rigidity or bending stiffness + ' + ' OUTPUT: + ' out - only for debugging messages + ' Pts - the list of points that approximate the shape of the elastica + ' Crv - the 3rd-degree curve interpolated from those points (with accurate start & end tangents) + ' L - the length of the rod/wire + ' W - the distance (width) between the endpoints of the rod/wire + ' H - the height of the bent rod/wire + ' A - the tangent angle at the (start) end of the rod/wire + ' F - the force needed to hold the rod/wire in a specific shape (based on the material properties & cross-section) **be sure your units for 'I' match your units for the + ' rest of your inputs (length, width, etc.). Also note that the critical buckling load (force) that makes the rod/wire start to bend can be found at height=0 + ' + ' THANKS TO: + ' MΓ₯rten Nettelbladt (thegeometryofbending.blogspot.com) + ' Daniel Piker (Kangaroo plugin) + ' David Rutten (Grasshopper guru) + ' Euler & Bernoulli (the O.G.'s) + ' + ' ----------------------------------------------------------------- + + Dim ignoreSelfIntersecting As Boolean = False ' set to True if you don't want to output curves where width < 0, which creates a self-intersecting curve + + Dim inCt As Integer = 0 ' count the number of required parameters that are receiving data + Dim length As Double + Dim width As System.Object = Nothing ' need to set as Nothing so we can check if it has been assigned a value later + Dim height As Double + Dim angle As Double + Dim m As Double + Dim multiple_m As New List(Of Double) + Dim AtoB As Line + Dim flip_H As Boolean = False ' if height is negative, this flag will be set + Dim flip_A As Boolean = False ' if angle is negative, this flag will be set + + If Not IsSet("Pln") Then + Msg("error", "Base plane is not set") + Return + End If + + If Not IsSet("PtA") Then + Msg("error", "Point A is not set") + Return + End If + + If Math.Round(Pln.DistanceTo(PtA), Defined.ROUNDTO) <> 0 Then + Msg("error", "Point A is not on the base plane") + Return + End If + + Dim refPlane As Plane = Pln ' create a reference plane = input plane and set the origin of it to PtA in case PtA isn't the origin already + refPlane.Origin = PtA + + If IsSet("PtB") Then + If Math.Round(Pln.DistanceTo(PtB), Defined.ROUNDTO) <> 0 Then + Msg("error", "Point B is not on the base plane") + Return + End If + + AtoB = New Line(PtA, PtB) + If AtoB.Length <> 0 And Not AtoB.Direction.IsPerpendicularTo(Pln.YAxis) Then + Msg("error", "The line between PtA and PtB is not perpendicular to the Y-axis of the specified plane") + Return + End If + + inCt += 1 + If IsSet("Wid") Then Msg("info", "Wid will override the distance between PtA and PtB. If you do not want this to happen, disconnect PtB or Wid.") + + width = PtA.DistanceTo(PtB) ' get the width (distance) between PtA and PtB + + Dim refPtB As Point3d + refPlane.RemapToPlaneSpace(PtB, refPtB) + If refPtB.X < 0 Then width = -width ' check if PtB is to the left of PtA...if so, width is negative + End If + + If IsSet("Len") Then inCt += 1 + If IsSet("Wid") Then inCt += 1 + If IsSet("Ht") Then inCt += 1 + If IsSet("Ang") Then inCt += 1 + If inCt > 2 Then Msg("info", "More parameters set than are required (out of length, width, height, angle). Only using the first two valid ones.") + + ' check for connected/specified inputs. note: only the first two that it comes across will be used + If IsSet("Len") Then ' if length is specified then... + If Len <= 0 Then + Msg("error", "Length cannot be negative or zero") + Return + End If + If IsSet("Wid") Then ' find height & angle based on length and specified width + If Wid > Len Then + Msg("error", "Width is greater than length") + Return + End If + If Wid = Len Then ' skip the solver and set the known values + height = 0 + m = 0 + angle = 0 + width = Wid + Else + m = SolveMFromLenWid(Len, Wid) + height = Cal_H(Len, m) ' L * Sqrt(m) / K(m) + angle = Cal_A(m) ' Acos(1 - 2 * m) + width = Wid + End If + + Else If width IsNot Nothing Then ' find height & angle based on length and calculated width (distance between PtA and PtB) + If width > Len Then + Msg("error", "Width is greater than length") + Return + End If + If width = Len Then ' skip the solver and set the known values + height = 0 + m = 0 + angle = 0 + Else + m = SolveMFromLenWid(Len, width) + height = Cal_H(Len, m) ' L * Sqrt(m) / K(m) + angle = Cal_A(m) ' Acos(1 - 2 * m) + End If + + Else If IsSet("Ht") Then ' find width & angle based on length and height ** possible to return 2 results ** + If Math.Abs(Ht / Len) > Defined.MAX_HL_RATIO Then + Msg("error", "Height not possible with given length") + Return + End If + If Ht < 0 Then + Ht = -Ht ' if height is negative, set it to positive (for the calculations) but flip the reference plane about its x-axis + refPlane.Transform(Transform.Mirror(New Plane(refPlane.Origin, refPlane.XAxis, refPlane.ZAxis))) + flip_A = True + flip_H = True + End If + If Ht = 0 Then ' skip the solver and set the known values + width = Len + angle = 0 + Else + multiple_m = SolveMFromLenHt(Len, Ht) ' note that it's possible for two values of m to be found if height is close to max height + If multiple_m.Count = 1 Then ' if there's only one m value returned, calculate the width & angle here. we'll deal with multiple m values later + m = multiple_m.Item(0) + width = Cal_W(Len, m) ' L * (2 * E(m) / K(m) - 1) + angle = Cal_A(m) ' Acos(1 - 2 * m) + End If + End If + height = Ht + + Else If IsSet("Ang") Then ' find width & height based on length and angle + If Ang < 0 Then + Ang = -Ang ' if angle is negative, set it to positive (for the calculations) but flip the reference plane about its x-axis + refPlane.Transform(Transform.Mirror(New Plane(refPlane.Origin, refPlane.XAxis, refPlane.ZAxis))) + flip_A = True + flip_H = True + End If + m = Cal_M(Ang) ' (1 - Cos(a)) / 2 + If Ang = 0 Then ' skip the solver and set the known values + width = Len + height = 0 + Else + width = Cal_W(Len, m) ' L * (2 * E(m) / K(m) - 1) + height = Cal_H(Len, m) ' L * Sqrt(m) / K(m) + End If + angle = Ang + + Else + Msg("error", "Need to specify one more parameter in addition to length") + Return + End If + length = Len + + Else If IsSet("Wid") Then ' if width is specified then... + If IsSet("Ht") Then ' find length & angle based on specified width and height + If Ht < 0 Then + Ht = -Ht ' if height is negative, set it to positive (for the calculations) but flip the reference plane about its x-axis + refPlane.Transform(Transform.Mirror(New Plane(refPlane.Origin, refPlane.XAxis, refPlane.ZAxis))) + flip_A = True + flip_H = True + End If + If Ht = 0 Then ' skip the solver and set the known values + length = Wid + angle = 0 + Else + m = SolveMFromWidHt(Wid, Ht) + length = Cal_L(Ht, m) ' h * K(m) / Sqrt(m) + angle = Cal_A(m) ' Acos(1 - 2 * m) + End If + height = Ht + + Else If IsSet("Ang") Then ' find length & height based on specified width and angle + If Wid = 0 Then + Msg("error", "Curve not possible with width = 0 and an angle as inputs") + Return + End If + If Ang < 0 Then + Ang = -Ang ' if angle is negative, set it to positive (for the calculations) but flip the reference plane about its x-axis + refPlane.Transform(Transform.Mirror(New Plane(refPlane.Origin, refPlane.XAxis, refPlane.ZAxis))) + flip_A = True + flip_H = True + End If + m = Cal_M(Ang) ' (1 - Cos(a)) / 2 + If Ang = 0 Then ' skip the solver and set the known values + length = Wid + height = 0 + Else + length = Wid / (2 * EllipticE(m) / EllipticK(m) - 1) + If length < 0 Then + Msg("error", "Curve not possible at specified width and angle (calculated length is negative)") + Return + End If + height = Cal_H(length, m) ' L * Sqrt(m) / K(m) + End If + angle = Ang + + Else + Msg("error", "Need to specify one more parameter in addition to width (Wid)") + Return + End If + width = Wid + + Else If width IsNot Nothing Then ' if width is determined by PtA and PtB then... + If IsSet("Ht") Then ' find length & angle based on calculated width and height + If Ht < 0 Then + Ht = -Ht ' if height is negative, set it to positive (for the calculations) but flip the reference plane about its x-axis + refPlane.Transform(Transform.Mirror(New Plane(refPlane.Origin, refPlane.XAxis, refPlane.ZAxis))) + flip_A = True + flip_H = True + End If + If Ht = 0 Then ' skip the solver and set the known values + length = width + angle = 0 + Else + m = SolveMFromWidHt(width, Ht) + length = Cal_L(Ht, m) ' h * K(m) / Sqrt(m) + angle = Cal_A(m) ' Acos(1 - 2 * m) + End If + height = Ht + + Else If IsSet("Ang") Then ' find length & height based on calculated width and angle + If width = 0 Then + Msg("error", "Curve not possible with width = 0 and an angle as inputs") + Return + End If + If Ang < 0 Then + Ang = -Ang ' if angle is negative, set it to positive (for the calculations) but flip the reference plane about its x-axis + refPlane.Transform(Transform.Mirror(New Plane(refPlane.Origin, refPlane.XAxis, refPlane.ZAxis))) + flip_A = True + flip_H = True + End If + m = Cal_M(Ang) ' (1 - Cos(a)) / 2 + If Ang = 0 Then ' skip the solver and set the known values + length = width + height = 0 + Else + length = width / (2 * EllipticE(m) / EllipticK(m) - 1) + If length < 0 Then + Msg("error", "Curve not possible at specified width and angle (calculated length is negative)") + Return + End If + height = Cal_H(length, m) ' L * Sqrt(m) / K(m) + End If + angle = Ang + + Else + Msg("error", "Need to specify one more parameter in addition to PtA and PtB") + Return + End If + + Else If IsSet("Ht") Then ' if height is specified then... + If IsSet("Ang") Then ' find length & width based on height and angle + If Ht < 0 Then + Ht = -Ht ' if height is negative, set it to positive (for the calculations) but flip the reference plane about its x-axis + refPlane.Transform(Transform.Mirror(New Plane(refPlane.Origin, refPlane.XAxis, refPlane.ZAxis))) + flip_H = True + flip_A = True + End If + If Ht = 0 Then + Msg("error", "Height can't = 0 if only height and angle are specified") + Return + Else + If Ang < 0 Then + Ang = -Ang ' if angle is negative, set it to positive (for the calculations) but flip the reference plane about its x-axis + refPlane.Transform(Transform.Mirror(New Plane(refPlane.Origin, refPlane.XAxis, refPlane.ZAxis))) + flip_A = Not flip_A + flip_H = Not flip_H + End If + m = Cal_M(Ang) ' (1 - Cos(a)) / 2 + If Ang = 0 Then + Msg("error", "Angle can't = 0 if only height and angle are specified") + Return + Else + length = Cal_L(Ht, m) ' h * K(m) / Sqrt(m) + width = Cal_W(length, m) ' L * (2 * E(m) / K(m) - 1) + End If + angle = Ang + End If + height = Ht + + Else + Msg("error", "Need to specify one more parameter in addition to height") + Return + End If + + Else If IsSet("Ang") Then + Msg("error", "Need to specify one more parameter in addition to angle") + Return + Else + Msg("error", "Need to specify two of the four parameters: length, width (or PtB), height, and angle") + Return + End If + + If m > Defined.M_MAX Then + Msg("error", "Form of curve not solvable with current algorithm and given inputs") + Return + End If + + refPlane.Origin = refPlane.PointAt(width / 2, 0, 0) ' adjust the origin of the reference plane so that the curve is centered about the y-axis (start of the curve is at x = -width/2) + + If multiple_m.Count > 1 Then ' if there is more than one m value returned, calculate the width, angle, and curve for each + Dim multi_pts As New DataTree(Of Point3d) + Dim multi_crv As New List(Of Curve) + Dim tmp_pts As New List(Of Point3d) + Dim multi_W, multi_A, multi_F As New List(Of Double) + Dim j As Integer = 0 ' used for creating a new branch (GH_Path) for storing pts which is itself a list of points + + For Each m_val As Double In multiple_m + width = Cal_W(length, m_val) 'length * (2 * EllipticE(m_val) / EllipticK(m_val) - 1) + + If width < 0 And ignoreSelfIntersecting Then + Msg("warning", "One curve is self-intersecting. To enable these, set ignoreSelfIntersecting to False") + Continue For + End If + + If m_val >= Defined.M_SKETCHY Then Msg("info", "Accuracy of the curve whose width = " & Math.Round(width, 4) & " is not guaranteed") + + angle = Cal_A(m_val) 'Math.Asin(2 * m_val - 1) + refPlane.Origin = refPlane.PointAt(width / 2, 0, 0) ' adjust the origin of the reference plane so that the curve is centered about the y-axis (start of the curve is at x = -width/2) + + tmp_pts = FindBendForm(length, width, m_val, angle, refPlane) + multi_pts.AddRange(tmp_pts, New GH_Path(j)) + multi_crv.Add(MakeCurve(tmp_pts, angle, refPlane)) + + multi_W.Add(width) + If flip_A Then angle = -angle + multi_A.Add(angle) + + E = E * 10 ^ 9 ' Young's modulus input E is in GPa, so we convert to Pa here (= N/m^2) + multi_F.Add(EllipticK(m_val) ^ 2 * E * I / length ^ 2) ' from reference {4} pg. 79 + + j += 1 + refPlane.Origin = PtA ' reset the reference plane origin to PtA for the next m_val + 'Print("length=" & length & ", width=" & width & ", height=" & height & ", angle=" & angle & ", m=" & m_val & ", k=" & Math.Sqrt(m_val) & ", w/L=" & width / length & ", h/L=" & height / length & ", w/h=" & width / height) + Next + + ' assign the outputs + Pts = multi_pts + Crv = multi_crv + L = length + W = multi_W + If flip_H Then height = -height + H = height + A = multi_A + F = multi_F + + Else ' only deal with the single m value + If m >= Defined.M_SKETCHY Then Msg("info", "Accuracy of the curve at these parameters is not guaranteed") + + If width < 0 And ignoreSelfIntersecting Then + Msg("error", "Curve is self-intersecting. To enable these, set ignoreSelfIntersecting to False") + Return + End If + + Pts = FindBendForm(length, width, m, angle, refPlane) + Crv = MakeCurve(pts, angle, refPlane) + L = length + W = width + If flip_H Then height = -height + H = height + If flip_A Then angle = -angle + A = angle + + E = E * 10 ^ 9 ' Young's modulus input E is in GPa, so we convert to Pa here (= N/m^2) + F = EllipticK(m) ^ 2 * E * I / length ^ 2 ' from reference {4} pg. 79. Note: the critical buckling (that makes the rod/wire start to bend) can be found at height=0 (width=length) + + 'height = Math.Sqrt(((2 * Len / 5) ^ 2 - ((Wid - Len / 5) / 2) ^ 2) ' quick approximation discovered by MΓ₯rten of 'Geometry of Bending' fame ( http://tiny.cc/it2pbx ) + 'width = (Len +/- 2 * Math.Sqrt(4 * Len ^ 2 - 25 * Ht ^ 2)) / 5 ' derived from above + 'length = (2 * Math.Sqrt(15 * Ht ^ 2 + 4 * Wid ^ 2) - Wid) / 3 ' derived from above + + 'Print("length=" & length & ", width=" & width & ", height=" & height & ", angle=" & angle & ", m=" & m & ", k=" & Math.Sqrt(m) & ", w/L=" & width / length & ", h/L=" & height / length & ", w/h=" & width / height) + End If + + + + + + + 612 + 233 + 84 + 184 + + + 654 + 325 + + + + + + 9 + 84fa917c-1ed8-4db3-8be1-7bdc4a6495a2 + 84fa917c-1ed8-4db3-8be1-7bdc4a6495a2 + 84fa917c-1ed8-4db3-8be1-7bdc4a6495a2 + 84fa917c-1ed8-4db3-8be1-7bdc4a6495a2 + 84fa917c-1ed8-4db3-8be1-7bdc4a6495a2 + 84fa917c-1ed8-4db3-8be1-7bdc4a6495a2 + 84fa917c-1ed8-4db3-8be1-7bdc4a6495a2 + 84fa917c-1ed8-4db3-8be1-7bdc4a6495a2 + 84fa917c-1ed8-4db3-8be1-7bdc4a6495a2 + 8 + 3ede854e-c753-40eb-84cb-b48008f14fd4 + 8ec86459-bf01-4409-baee-174d0d2b13d0 + 8ec86459-bf01-4409-baee-174d0d2b13d0 + 8ec86459-bf01-4409-baee-174d0d2b13d0 + 8ec86459-bf01-4409-baee-174d0d2b13d0 + 8ec86459-bf01-4409-baee-174d0d2b13d0 + 8ec86459-bf01-4409-baee-174d0d2b13d0 + 8ec86459-bf01-4409-baee-174d0d2b13d0 + + + + + true + Script Variable PtA + 920df659-6d29-453d-9295-577245828ba6 + PtA + PtA + true + 0 + true + 7451bc70-5fc3-43a3-bb48-ff10952414e7 + 1 + e1937b56-b1da-4c12-8bd8-e34ee81746ef + + + + + + 614 + 235 + 25 + 20 + + + 628 + 245 + + + + + + + + true + Script Variable PtB + eeb8ccaa-8966-4eab-8949-3eb384a12d84 + PtB + PtB + true + 0 + true + d5104343-e872-4369-9a14-a75a852c1a15 + 1 + e1937b56-b1da-4c12-8bd8-e34ee81746ef + + + + + + 614 + 255 + 25 + 20 + + + 628 + 265 + + + + + + + + true + Script Variable Pln + ee4f4d3f-f195-437b-88af-35d3a73d66ad + Pln + Pln + true + 0 + true + df7d1e6a-049f-4594-9fb2-7dda33d26e57 + 1 + 3897522d-58e9-4d60-b38c-978ddacfedd8 + + + + + + 614 + 275 + 25 + 20 + + + 628 + 285 + + + + + + + + true + Script Variable Len + 999531d8-8fc7-4421-8afb-076eb4ce3f6e + Len + Len + true + 0 + true + ce2f14ec-483c-4899-a8cb-784a62168957 + 1 + 8e991e99-5fb8-41e1-928d-1bba8fb9f7d7 + + + + + + 614 + 295 + 25 + 20 + + + 628 + 305 + + + + + + + + true + Script Variable Wid + 8aba3acb-d87c-46a0-aef3-179156140406 + Wid + Wid + true + 0 + true + 0 + 8e991e99-5fb8-41e1-928d-1bba8fb9f7d7 + + + + + + 614 + 315 + 25 + 20 + + + 628 + 325 + + + + + + + + true + Script Variable Ht + 54082c0a-ad9c-49e6-97c2-34b9d8c0e605 + Ht + Ht + true + 0 + true + 0 + 8e991e99-5fb8-41e1-928d-1bba8fb9f7d7 + + + + + + 614 + 335 + 25 + 20 + + + 628 + 345 + + + + + + + + true + Script Variable Ang + 998111e9-4c7d-4b27-88a9-01982081691a + Ang + Ang + true + 0 + true + 0 + 8e991e99-5fb8-41e1-928d-1bba8fb9f7d7 + + + + + + 614 + 355 + 25 + 20 + + + 628 + 365 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681.5 + 313.75 + + + + + + + + Output parameter W + 816f49f4-39a6-4705-80c7-e2a924ac1e0c + W + W + false + 0 + + + + + + 669 + 325 + 25 + 22 + + + 681.5 + 336.25 + + + + + + + + Output parameter H + b00909dc-b385-4e3e-a3a8-9e76efdaadeb + H + H + false + 0 + + + + + + 669 + 347 + 25 + 23 + + + 681.5 + 358.75 + + + + + + + + Output parameter A + 9632d9b7-ad5c-4b42-bc41-5bf9a4af0115 + A + A + false + 0 + + + + + + 669 + 370 + 25 + 22 + + + 681.5 + 381.25 + + + + + + + + Output parameter F + 28c91c87-29e8-4bc8-a5ff-18aadc6f0ecd + F + F + false + 0 + + + + + + 669 + 392 + 25 + 23 + + + 681.5 + 403.75 + + + + + + + + + + + + + + 57da07bd-ecab-415d-9d86-af36d7073abc + Number Slider + + + + + Numeric slider for single values + 19d4e5e6-a3fb-4e4d-b426-93c0b41f974c + Number Slider + width + false + 0 + + + + + + 158 + 312 + 384 + 20 + + + 158.3465 + 312.3785 + + + + + + 2 + 1 + 0 + 400 + -130 + 0 + 183.21 + + + + + + + + + 57da07bd-ecab-415d-9d86-af36d7073abc + Number Slider + + + + + Numeric slider for single values + ce2f14ec-483c-4899-a8cb-784a62168957 + Number Slider + length + false + 0 + + + + + + 158 + 285 + 385 + 20 + + + 158.0028 + 285.5286 + + + + + + 2 + 1 + 0 + 400 + 0 + 0 + 300 + + + + + + + + + fbac3e32-f100-4292-8692-77240a42fd1a + Point + + + + + Contains a collection of three-dimensional points + true + b2a67d0f-c66e-46a9-8efd-f7442d233d5d + Point + Pt + false + d4f9a77d-b3a2-46c6-91de-6ce275666f2d + 1 + + + + + + 782 + 194 + 50 + 24 + + + 807.4574 + 206.2478 + + + + + + + + + + 59e0b89a-e487-49f8-bab8-b5bab16be14c + Panel + + + + + A panel for custom notes and text values + 32bb1a9f-9575-4b8c-8a60-a65a7b9dd15f + Panel + + false + 0 + 2296b093-286c-438d-aa59-465d23147f1c + 1 + Double click to edit panel content… + + + + + + 855 + 408 + 105 + 55 + + 0 + 0 + 0 + + 855.6731 + 408.6088 + + + + + + + 255;255;250;90 + + true + true + true + false + false + true + + + + + + + + + 0d77c51e-584f-44e8-aed2-c2ddf4803888 + Degrees + + + + + Convert an angle specified in radians to degrees + 98102773-859e-4cf3-83a5-41f68379af66 + Degrees + Deg + + + + + + 754 + 421 + 64 + 28 + + + 784 + 435 + + + + + + Angle in radians + f013de98-8461-42d6-94e2-d4f473814c3f + Radians + R + false + 9632d9b7-ad5c-4b42-bc41-5bf9a4af0115 + 1 + + + + + + 756 + 423 + 13 + 24 + + + 764 + 435 + + + + + + + + Angle in degrees + 2296b093-286c-438d-aa59-465d23147f1c + Degrees + D + false + 0 + + + + + + 799 + 423 + 17 + 24 + + + 807.5 + 435 + + + + + + + + + + + + 3581f42a-9592-4549-bd6b-1c0fc39d067b + Construct Point + + + + + Construct a point from {xyz} coordinates. + d68f5884-1ed1-4bd5-ab64-b7040370d59b + Construct Point + Pt + + + + + + 392 + 101 + 67 + 64 + + + 423 + 133 + + + + + + {x} coordinate + b9e9716b-aaed-4e63-90f0-fd69bffec388 + X coordinate + X + false + 0 + + + + + + 394 + 103 + 14 + 20 + + + 402.5 + 113 + + + + + + 1 + + + + + 1 + {0} + + + + + 0 + + + + + + + + + + + {y} coordinate + 961e58c0-1cd5-49a0-8fd1-0e419a2c0b34 + Y coordinate + Y + false + 0 + + + + + + 394 + 123 + 14 + 20 + + + 402.5 + 133 + + + + + + 1 + + + + + 1 + {0} + + + + + 0 + + + + + + + + + + + {z} coordinate + 54bdcf81-cd9a-447c-bad3-a55fe7ef5dc1 + Z coordinate + Z + false + 0 + + + + + + 394 + 143 + 14 + 20 + + + 402.5 + 153 + + + + + + 1 + + + + + 1 + {0} + + + + + 0 + + + + + + + + + + + Point coordinate + 7451bc70-5fc3-43a3-bb48-ff10952414e7 + Point + Pt + false + 0 + + + + + + 438 + 103 + 19 + 60 + + + 447.5 + 133 + + + + + + + + + + + + 3581f42a-9592-4549-bd6b-1c0fc39d067b + Construct Point + + + + + Construct a point from {xyz} coordinates. + 8cd6ad76-7f71-4948-8c5e-9a3e2549985f + Construct Point + Pt + + + + + + 392 + 173 + 67 + 64 + + + 423 + 205 + + + + + + {x} coordinate + cc48fd1f-8953-40bd-a5f6-a9a203bcabd4 + X coordinate + X + false + 95f9fd7f-37dc-4bd8-8105-7301ef052bdd + 1 + + + + + + 394 + 175 + 14 + 20 + + + 402.5 + 185 + + + + + + 1 + + + + + 1 + {0} + + + + + 80 + + + + + + + + + + + {y} coordinate + a1e74152-bea6-4daf-a4cf-bbaa027d9769 + Y coordinate + Y + false + 0 + + + + + + 394 + 195 + 14 + 20 + + + 402.5 + 205 + + + + + + 1 + + + + + 1 + {0} + + + + + 0 + + + + + + + + + + + {z} coordinate + e34fa320-f5a3-4aaf-beaf-4207678e6e88 + Z coordinate + Z + false + 0 + + + + + + 394 + 215 + 14 + 20 + + + 402.5 + 225 + + + + + + 1 + + + + + 1 + {0} + + + + + 0 + + + + + + + + + + + Point coordinate + d5104343-e872-4369-9a14-a75a852c1a15 + Point + Pt + false + 0 + + + + + + 438 + 175 + 19 + 60 + + + 447.5 + 205 + + + + + + + + + + + + d5967b9f-e8ee-436b-a8ad-29fdcecf32d5 + Curve + + + + + Contains a collection of generic curves + d53a1087-053a-44d5-b485-68a8b5d09ce4 + Curve + Crv + false + 1b8d948b-eedb-4c3c-bfba-ceaee74ff110 + 1 + + + + + + 782 + 236 + 50 + 24 + + + 807.4463 + 248.7776 + + + + + + + + + + 17b7152b-d30d-4d50-b9ef-c9fe25576fc2 + XY Plane + + + + + World XY plane. + true + cc8dfb80-5022-4b13-83c9-a787888900e8 + XY Plane + XY + + + + + + 474 + 246 + 64 + 28 + + + 505 + 260 + + + + + + Origin of plane + 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"warning" + Component.AddRuntimeMessage(GH_RuntimeMessageLevel.Warning, msg) + Print("Warning: " & msg) + Case "info" + Component.AddRuntimeMessage(GH_RuntimeMessageLevel.Remark, msg) + Print(msg) + End Select + End Sub + + ' Solve for the m parameter from length and width (reference {1} equation (34), except b = width and K(k) and E(k) should be K(m) and E(m)) + Private Function SolveMFromLenWid(ByVal L As Double, ByVal w As Double) As Double + If w = 0 Then + Return Defined.M_ZERO_W ' for the boundry condition width = 0, bypass the function and return the known m value + End If + + Dim n As Integer = 1 ' Iteration counter (quit if >MAXIT) + Dim lower As Double = 0 ' m must be within this range + Dim upper As Double = 1 + Dim m As Double + Dim cwl As Double + + Do While (upper - lower) > Defined.MAXERR AndAlso (n) < Defined.MAXIT ' Repeat until range narrow enough or MAXIT + m = (upper + lower) / 2 + cwl = 2 * EllipticE(m) / EllipticK(m) - 1 ' calculate w/L with the test value of m + If cwl < w / L Then ' compares the calculated w/L with the actual w/L then narrows the range of possible m + upper = m + Else + lower = m + End If + n += 1 + Loop + Return m + End Function + + ' Solve for the m parameter from length and height (reference {1} equation (33), except K(k) should be K(m) and k = sqrt(m)) + ' Note that it's actually possible to find 2 valid values for m (hence 2 width values) at certain height values + Private Function SolveMFromLenHt(ByVal L As Double, ByVal h As Double) As List(Of Double) + Dim n As Integer = 1 ' Iteration counter (quit if >MAXIT) + Dim lower As Double = 0 ' m must be within this range + Dim upper As Double = 1 + Dim twoWidths As Boolean = h / L >= Defined.DOUBLE_W_HL_RATIO And h / L < Defined.MAX_HL_RATIO ' check to see if h/L is within the range where 2 solutions for the width are possible + Dim m As Double + Dim mult_m As New List(Of Double) + Dim chl As Double + + If twoWidths Then + ' find the first of two possible solutions for m with the following limits: + lower = Defined.M_DOUBLE_W ' see constants at bottom of script + upper = Defined.M_MAXHEIGHT ' see constants at bottom of script + Do While (upper - lower) > Defined.MAXERR AndAlso (n) < Defined.MAXIT ' Repeat until range narrow enough or MAXIT + m = (upper + lower) / 2 + chl = Math.Sqrt(m) / EllipticK(m) ' calculate h/L with the test value of m + If chl > h / L Then ' compares the calculated h/L with the actual h/L then narrows the range of possible m + upper = m + Else + lower = m + End If + n += 1 + Loop + mult_m.Add(m) + + ' then find the second of two possible solutions for m with the following limits: + lower = Defined.M_MAXHEIGHT ' see constants at bottom of script + upper = 1 + Do While (upper - lower) > Defined.MAXERR AndAlso (n) < Defined.MAXIT ' Repeat until range narrow enough or MAXIT + m = (upper + lower) / 2 + chl = Math.Sqrt(m) / EllipticK(m) ' calculate h/L with the test value of m + If chl < h / L Then ' compares the calculated h/L with the actual h/L then narrows the range of possible m + upper = m + Else + lower = m + End If + n += 1 + Loop + + If m <= Defined.M_MAX Then ' return this m parameter only if it falls within the maximum useful value (above which the curve breaks down) + mult_m.Add(m) + End If + + Else + ' find the one possible solution for the m parameter + upper = Defined.M_DOUBLE_W ' limit the upper end of the search to the maximum value of m for which only one solution exists + Do While (upper - lower) > Defined.MAXERR AndAlso (n) < Defined.MAXIT ' Repeat until range narrow enough or MAXIT + m = (upper + lower) / 2 + chl = Math.Sqrt(m) / EllipticK(m) ' calculate h/L with the test value of m + If chl > h / L Then ' compares the calculated h/L with the actual h/L then narrows the range of possible m + upper = m + Else + lower = m + End If + n += 1 + Loop + mult_m.Add(m) + End If + + Return mult_m + End Function + + ' Solve for the m parameter from width and height (derived from reference {1} equations (33) and (34) with same notes as above) + Private Function SolveMFromWidHt(ByVal w As Double, ByVal h As Double) As Double + Dim n As Integer = 1 ' Iteration counter (quit if >MAXIT) + Dim lower As Double = 0 ' m must be within this range + Dim upper As Double = 1 + Dim m As Double + Dim cwh As Double + + Do While (upper - lower) > Defined.MAXERR AndAlso (n) < Defined.MAXIT ' Repeat until range narrow enough or MAXIT + m = (upper + lower) / 2 + cwh = (2 * EllipticE(m) - EllipticK(m)) / Math.Sqrt(m) ' calculate w/h with the test value of m + If cwh < w / h Then ' compares the calculated w/h with the actual w/h then narrows the range of possible m + upper = m + Else + lower = m + End If + n += 1 + Loop + + Return m + End Function + + ' Calculate length based on height and an m parameter, derived from reference {1} equation (33), except K(k) should be K(m) and k = sqrt(m) + Private Function Cal_L(ByVal h As Double, ByVal m As Double) As Double + Return h * EllipticK(m) / Math.Sqrt(m) + End Function + + ' Calculate width based on length and an m parameter, derived from reference {1} equation (34), except b = width and K(k) and E(k) should be K(m) and E(m) + Private Function Cal_W(ByVal L As Double, ByVal m As Double) As Double + Return L * (2 * EllipticE(m) / EllipticK(m) - 1) + End Function + + ' Calculate height based on length and an m parameter, from reference {1} equation (33), except K(k) should be K(m) and k = sqrt(m) + Private Function Cal_H(ByVal L As Double, ByVal m As Double) As Double + Return L * Math.Sqrt(m) / EllipticK(m) + End Function + + ' Calculate the unique m parameter based on a start tangent angle, from reference {2}, just above equation (9a), that states k = Sin(angle / 2 + Pi / 4), + ' but as m = k^2 and due to this script's need for an angle rotated 90Β° versus the one in reference {1}, the following formula is the result + ' New note: verified by reference {4}, pg. 78 at the bottom + Private Function Cal_M(ByVal a As Double) As Double + Return (1 - Math.Cos(a)) / 2 ' equal to Sin^2(a/2) too + End Function + + ' Calculate start tangent angle based on an m parameter, derived from above formula + Private Function Cal_A(ByVal m As Double) As Double + Return Math.Acos(1 - 2 * m) + End Function + + ' This is the heart of this script, taking the found (or specified) length, width, and angle values along with the found m parameter to create + ' a list of points that approximate the shape or form of the elastica. It works by finding the x and y coordinates (which are reversed versus + ' the original equations (12a) and (12b) from reference {2} due to the 90Β° difference in orientation) based on the tangent angle along the curve. + ' See reference {2} for more details on how they derived it. Note that to simplify things, the algorithm only calculates the points for half of the + ' curve, then mirrors those points along the y-axis. + Private Function FindBendForm(ByVal L As Double, ByVal w As Double, ByVal m As Double, ByVal ang As Double, ByVal refPln As Plane) As List(Of Point3d) + L = L / 2 ' because the below algorithm is based on the formulas in reference {2} for only half of the curve + w = w / 2 ' same + + If ang = 0 Then ' if angle (and height) = 0, then simply return the start and end points of the straight line + Dim out As New List(Of Point3d) + out.Add(refPln.PointAt(w, 0, 0)) + out.Add(refPln.PointAt(-w, 0, 0)) + Return out + End If + + Dim x As Double + Dim y As Double + Dim halfCurvePts As New List(Of Point3d) + Dim fullCurvePts As New List(Of Point3d) + Dim translatedPts As New List(Of Point3d) + + ang -= Math.PI / 2 ' a hack to allow this algorithm to work, since the original curve in paper {2} was rotated 90Β° + Dim angB As Double = ang + (-Math.PI / 2 - ang) / Defined.CURVEDIVS ' angB is the 'lowercase theta' which should be in formula {2}(12b) as the interval + ' start [a typo...see equation(3)]. It's necessary to start angB at ang + [interval] instead of just ang due to integration failing at angB = ang + halfCurvePts.Add(New Point3d(w, 0, 0)) ' start with this known initial point, as integration will fail when angB = ang + + ' each point {x, y} is calculated from the tangent angle, angB, that occurs at each point (which is why this iterates from ~ang to -pi/2, the known end condition) + Do While Math.Round(angB, Defined.ROUNDTO) >= Math.Round(-Math.PI / 2, Defined.ROUNDTO) + y = (Math.Sqrt(2) * Math.Sqrt(Math.Sin(ang) - Math.Sin(angB)) * (w + L)) / (2 * EllipticE(m)) ' note that x and y are swapped vs. (12a) and (12b) + x = (L / (Math.Sqrt(2) * EllipticK(m))) * Simpson(angB, -Math.PI / 2, 500, ang) ' calculate the Simpson approximation of the integral (function f below) + ' over the interval angB ('lowercase theta') to -pi/2. side note: is 500 too few iterations for the Simson algorithm? + + If Math.Round(x, Defined.ROUNDTO) = 0 Then x = 0 + halfCurvePts.Add(New Point3d(x, y, 0)) + + angB += (-Math.PI / 2 - ang) / Defined.CURVEDIVS ' onto the next tangent angle + Loop + + ' After finding the x and y values for half of the curve, add the {-x, y} values for the rest of the curve + For Each point As Point3d In halfCurvePts + If Math.Round(point.X, Defined.ROUNDTO) = 0 Then + If Math.Round(point.Y, Defined.ROUNDTO) = 0 Then + fullCurvePts.Add(New Point3d(0, 0, 0)) ' special case when width = 0: when x = 0, only duplicate the point when y = 0 too + End If + Else + fullCurvePts.Add(New Point3d(-point.X, point.Y, 0)) + End If + Next + halfCurvePts.Reverse + fullCurvePts.AddRange(halfCurvePts) + + For Each p As Point3d In fullCurvePts + translatedPts.Add(refPln.PointAt(p.X, p.Y, p.Z)) ' translate the points from the reference plane to the world plane + Next + + Return translatedPts + End Function + + ' Interpolates the points from FindBendForm to create the Elastica curve. Uses start & end tangents for greater accuracy. + Private Function MakeCurve(ByVal pts As List(Of Point3d), ByVal ang As Double, ByVal refPln As Plane) As Curve + If ang <> 0 Then + Dim ts, te As New Vector3d(refPln.XAxis) + ts.Rotate(ang, refPln.ZAxis) + te.Rotate(-ang, refPln.ZAxis) + Return Curve.CreateInterpolatedCurve(pts, 3, CurveKnotStyle.Chord, ts, te) ' 3rd degree curve with 'Chord' Knot Style + Else + Return Curve.CreateInterpolatedCurve(pts, 3) ' if angle (and height) = 0, then simply interpolate the straight line (no start/end tangents) + End If + End Function + + ' Implements the Simpson approximation for an integral of function f below + Public Function Simpson(a As Double, b As Double, n As Integer, theta As Double) As Double 'n should be an even number + Dim j As Integer, s1 As Double, s2 As Double, h As Double + h = (b - a) / n + s1 = 0 + s2 = 0 + For j = 1 To n - 1 Step 2 + s1 = s1 + fn(a + j * h, theta) + Next j + For j = 2 To n - 2 Step 2 + s2 = s2 + fn(a + j * h, theta) + Next j + Simpson = h / 3 * (fn(a, theta) + 4 * s1 + 2 * s2 + fn(b, theta)) + End Function + + ' Specific calculation for the above integration + Public Function fn(x As Double, theta As Double) As Double + fn = Math.Sin(x) / (Math.Sqrt(Math.Sin(theta) - Math.Sin(x))) ' from reference {2} formula (12b) + End Function + + + ' Return the Complete Elliptic integral of the 1st kind + ' Abramowitz and Stegun p.591, formula 17.3.11 + ' Code from http://www.codeproject.com/Articles/566614/Elliptic-integrals + Public Function EllipticK(ByVal m As Double) As Double + Dim sum, term, above, below As Double + sum = 1 + term = 1 + above = 1 + below = 2 + + For i As Integer = 1 To 100 + term *= above / below + sum += Math.Pow(m, i) * Math.Pow(term, 2) + above += 2 + below += 2 + Next + sum *= 0.5 * Math.PI + Return sum + End Function + + + ' Return the Complete Elliptic integral of the 2nd kind + ' Abramowitz and Stegun p.591, formula 17.3.12 + ' Code from http://www.codeproject.com/Articles/566614/Elliptic-integrals + Public Function EllipticE(ByVal m As Double) As Double + Dim sum, term, above, below As Double + sum = 1 + term = 1 + above = 1 + below = 2 + + For i As Integer = 1 To 100 + term *= above / below + sum -= Math.Pow(m, i) * Math.Pow(term, 2) / above + above += 2 + below += 2 + Next + sum *= 0.5 * Math.PI + Return sum + End Function + + Friend Partial NotInheritable Class Defined + Private Sub New() + End Sub + + ' Note: most of these values for m and h/L ratio were found with Wolfram Alpha and either specific intercepts (x=0) or local minima/maxima. They should be constant. + Public Const M_SKETCHY As Double = 0.95 ' value of the m parameter where the curvature near the ends of the curve gets wonky + Public Const M_MAX As Double = 0.993 ' maximum useful value of the m parameter, above which this algorithm for the form of the curve breaks down + Public Const M_ZERO_W As Double = 0.826114765984970336 ' value of the m parameter when width = 0 + Public Const M_MAXHEIGHT As Double = 0.701327460663101223 ' value of the m parameter at maximum possible height of the bent rod/wire + Public Const M_DOUBLE_W As Double = 0.180254422335013983 ' minimum value of the m parameter when two width values are possible for a given height and length + Public Const DOUBLE_W_HL_RATIO As Double = 0.257342117984635757 ' value of the height/length ratio above which there are two possible width values + Public Const MAX_HL_RATIO As Double = 0.403140189705650243 ' maximum possible value of the height/length ratio + + Public Const MAXERR As Double = 0.0000000001 ' error tolerance + Public Const MAXIT As Integer = 100 ' maximum number of iterations + Public Const ROUNDTO As Integer = 10 ' number of decimal places to round off to + Public Const CURVEDIVS As Integer = 50 ' number of sample points for building the curve (or half-curve as it were) + End Class + A VB.NET scriptable component + + 98 + 86 + + true + 34e9c6ff-5f0a-453a-89bb-504c40c19604 + VB Script + VB + true + 0 + ' ----------------------------------------------------------------- + ' Elastic Bending Script by Will McElwain + ' Created February 2014 + ' + ' DESCRIPTION: + ' This beast creates the so-called 'elastica curve', the shape a long, thin rod or wire makes when it is bent elastically (i.e. not permanently). In this case, force + ' is assumed to only be applied horizontally (which would be in line with the rod at rest) and both ends are assumed to be pinned or hinged meaning they are free + ' to rotate (as opposed to clamped, when the end tangent angle is fixed, usually horizontally). An interesting finding is that it doesn't matter what the material or + ' cross-sectional area is, as long as they're uniform along the entire length. Everything makes the same shape when bent as long as it doesn't cross the threshold + ' from elastic to plastic (permanent) deformation (I don't bother to find that limit here, but can be found if the yield stress for a material is known). + ' + ' Key to the formulas used in this script are elliptic integrals, specifically K(m), the complete elliptic integral of the first kind, and E(m), the complete elliptic + ' integral of the second kind. There was a lot of confusion over the 'm' and 'k' parameters for these functions, as some people use them interchangeably, but they are + ' not the same. m = k^2 (thus k = Sqrt(m)). I try to use the 'm' parameter exclusively to avoid this confusion. Note that there is a unique 'm' parameter for every + ' configuration/shape of the elastica curve. + ' + ' This script tries to find that unique 'm' parameter based on the inputs. The algorithm starts with a test version of m, evaluates an expression, say 2*E(m)/K(m)-1, + ' then compares the result to what it should be (in this case, a known width/length ratio). Iterate until the correct m is found. Once we have m, we can then calculate + ' all of the other unknowns, then find points that lie on that curve, then interpolate those points for the actual curve. You can also use Wolfram|Alpha as I did to + ' find the m parameter based on the equations in this script (example here: http://tiny.cc/t4tpbx for when say width=45.2 and length=67.1). + ' + ' Other notes: + ' * This script works with negative values for width, which will creat a self-intersecting curve (as it should). The curvature of the elastica starts to break down around + ' m=0.95 (~154Β°), but this script will continue to work until M_MAX, m=0.993 (~169Β°). If you wish to ignore self-intersecting curves, set ignoreSelfIntersecting to True + ' * When the only known values are length and height, it is actually possible for certain ratios of height to length to have two valid m values (thus 2 possible widths + ' and angles). This script will return them both. + ' * Only the first two valid parameters (of the required ones) will be used, meaning if all four are connected (length, width or a PtB, height, and angle), this script will + ' only use length and width (or a PtB). + ' * Depending on the magnitude of your inputs (say if they're really small, like if length < 10), you might have to increase the constant ROUNDTO at the bottom + ' + ' REFERENCES: + ' {1} "The elastic rod" by M.E. Pacheco Q. & E. Pina, http://www.scielo.org.mx/pdf/rmfe/v53n2/v53n2a8.pdf + ' {2} "An experiment in nonlinear beam theory" by A. Valiente, http://www.deepdyve.com/lp/doc/I3lwnxdfGz , also here: http://tiny.cc/Valiente_AEiNBT + ' {3} "Snap buckling, writhing and Loop formation In twisted rods" by V.G.A. GOSS, http://myweb.lsbu.ac.uk/~gossga/thesisFinal.pdf + ' {4} "Theory of Elastic Stability" by Stephen Timoshenko, http://www.scribd.com/doc/50402462/Timoshenko-Theory-of-Elastic-Stability (start on p. 76) + ' + ' INPUT: + ' PtA - First anchor point (required) + ' PtB - Second anchor point (optional, though 2 out of the 4--length, width, height, angle--need to be specified) + ' [note that PtB can be the same as PtA (meaning width would be zero)] + ' [also note that if a different width is additionally specified that's not equal to the distance between PtA and PtB, then the end point will not equal PtB anymore] + ' Pln - Plane of the bent rod/wire, which bends up in the +y direction. The line between PtA and PtB (if specified) must be parallel to the x-axis of this plane + ' + ' ** 2 of the following 4 need to be specified ** + ' Len - Length of the rod/wire, which needs to be > 0 + ' Wid - Width between the endpoints of the curve [note: if PtB is specified in addition, and distance between PtA and PtB <> width, the end point will be relocated + ' Ht - Height of the bent rod/wire (when negative, curve will bend downward, relative to the input plane, instead) + ' Ang - Inner departure angle or tangent angle (in radians) at the ends of the bent rod/wire. Set up so as width approaches length (thus height approaches zero), angle approaches zero + ' + ' * Following variables only needed for optional calculating of bending force, not for shape of curve. + ' E - Young's modulus (modulus of elasticity) in GPa (=N/m^2) (material-specific. for example, 7075 aluminum is roughly 71.7 GPa) + ' I - Second moment of area (or area moment of inertia) in m^4 (cross-section-specific. for example, a hollow rod + ' would have I = pi * (outer_diameter^4 - inner_diameter^4) / 32 + ' Note: E*I is also known as flexural rigidity or bending stiffness + ' + ' OUTPUT: + ' out - only for debugging messages + ' Pts - the list of points that approximate the shape of the elastica + ' Crv - the 3rd-degree curve interpolated from those points (with accurate start & end tangents) + ' L - the length of the rod/wire + ' W - the distance (width) between the endpoints of the rod/wire + ' H - the height of the bent rod/wire + ' A - the tangent angle at the (start) end of the rod/wire + ' F - the force needed to hold the rod/wire in a specific shape (based on the material properties & cross-section) **be sure your units for 'I' match your units for the + ' rest of your inputs (length, width, etc.). Also note that the critical buckling load (force) that makes the rod/wire start to bend can be found at height=0 + ' + ' THANKS TO: + ' MΓ₯rten Nettelbladt (thegeometryofbending.blogspot.com) + ' Daniel Piker (Kangaroo plugin) + ' David Rutten (Grasshopper guru) + ' Euler & Bernoulli (the O.G.'s) + ' + ' ----------------------------------------------------------------- + + Dim ignoreSelfIntersecting As Boolean = False ' set to True if you don't want to output curves where width < 0, which creates a self-intersecting curve + + Dim inCt As Integer = 0 ' count the number of required parameters that are receiving data + Dim length As Double + Dim width As System.Object = Nothing ' need to set as Nothing so we can check if it has been assigned a value later + Dim height As Double + Dim angle As Double + Dim m As Double + Dim multiple_m As New List(Of Double) + Dim AtoB As Line + Dim flip_H As Boolean = False ' if height is negative, this flag will be set + Dim flip_A As Boolean = False ' if angle is negative, this flag will be set + + If Not IsSet("Pln") Then + Msg("error", "Base plane is not set") + Return + End If + + If Not IsSet("PtA") Then + Msg("error", "Point A is not set") + Return + End If + + If Math.Round(Pln.DistanceTo(PtA), Defined.ROUNDTO) <> 0 Then + Msg("error", "Point A is not on the base plane") + Return + End If + + Dim refPlane As Plane = Pln ' create a reference plane = input plane and set the origin of it to PtA in case PtA isn't the origin already + refPlane.Origin = PtA + + If IsSet("PtB") Then + If Math.Round(Pln.DistanceTo(PtB), Defined.ROUNDTO) <> 0 Then + Msg("error", "Point B is not on the base plane") + Return + End If + + AtoB = New Line(PtA, PtB) + If AtoB.Length <> 0 And Not AtoB.Direction.IsPerpendicularTo(Pln.YAxis) Then + Msg("error", "The line between PtA and PtB is not perpendicular to the Y-axis of the specified plane") + Return + End If + + inCt += 1 + If IsSet("Wid") Then Msg("info", "Wid will override the distance between PtA and PtB. If you do not want this to happen, disconnect PtB or Wid.") + + width = PtA.DistanceTo(PtB) ' get the width (distance) between PtA and PtB + + Dim refPtB As Point3d + refPlane.RemapToPlaneSpace(PtB, refPtB) + If refPtB.X < 0 Then width = -width ' check if PtB is to the left of PtA...if so, width is negative + End If + + If IsSet("Len") Then inCt += 1 + If IsSet("Wid") Then inCt += 1 + If IsSet("Ht") Then inCt += 1 + If IsSet("Ang") Then inCt += 1 + If inCt > 2 Then Msg("info", "More parameters set than are required (out of length, width, height, angle). Only using the first two valid ones.") + + ' check for connected/specified inputs. note: only the first two that it comes across will be used + If IsSet("Len") Then ' if length is specified then... + If Len <= 0 Then + Msg("error", "Length cannot be negative or zero") + Return + End If + If IsSet("Wid") Then ' find height & angle based on length and specified width + If Wid > Len Then + Msg("error", "Width is greater than length") + Return + End If + If Wid = Len Then ' skip the solver and set the known values + height = 0 + m = 0 + angle = 0 + width = Wid + Else + m = SolveMFromLenWid(Len, Wid) + height = Cal_H(Len, m) ' L * Sqrt(m) / K(m) + angle = Cal_A(m) ' Acos(1 - 2 * m) + width = Wid + End If + + Else If width IsNot Nothing Then ' find height & angle based on length and calculated width (distance between PtA and PtB) + If width > Len Then + Msg("error", "Width is greater than length") + Return + End If + If width = Len Then ' skip the solver and set the known values + height = 0 + m = 0 + angle = 0 + Else + m = SolveMFromLenWid(Len, width) + height = Cal_H(Len, m) ' L * Sqrt(m) / K(m) + angle = Cal_A(m) ' Acos(1 - 2 * m) + End If + + Else If IsSet("Ht") Then ' find width & angle based on length and height ** possible to return 2 results ** + If Math.Abs(Ht / Len) > Defined.MAX_HL_RATIO Then + Msg("error", "Height not possible with given length") + Return + End If + If Ht < 0 Then + Ht = -Ht ' if height is negative, set it to positive (for the calculations) but flip the reference plane about its x-axis + refPlane.Transform(Transform.Mirror(New Plane(refPlane.Origin, refPlane.XAxis, refPlane.ZAxis))) + flip_A = True + flip_H = True + End If + If Ht = 0 Then ' skip the solver and set the known values + width = Len + angle = 0 + Else + multiple_m = SolveMFromLenHt(Len, Ht) ' note that it's possible for two values of m to be found if height is close to max height + If multiple_m.Count = 1 Then ' if there's only one m value returned, calculate the width & angle here. we'll deal with multiple m values later + m = multiple_m.Item(0) + width = Cal_W(Len, m) ' L * (2 * E(m) / K(m) - 1) + angle = Cal_A(m) ' Acos(1 - 2 * m) + End If + End If + height = Ht + + Else If IsSet("Ang") Then ' find width & height based on length and angle + If Ang < 0 Then + Ang = -Ang ' if angle is negative, set it to positive (for the calculations) but flip the reference plane about its x-axis + refPlane.Transform(Transform.Mirror(New Plane(refPlane.Origin, refPlane.XAxis, refPlane.ZAxis))) + flip_A = True + flip_H = True + End If + m = Cal_M(Ang) ' (1 - Cos(a)) / 2 + If Ang = 0 Then ' skip the solver and set the known values + width = Len + height = 0 + Else + width = Cal_W(Len, m) ' L * (2 * E(m) / K(m) - 1) + height = Cal_H(Len, m) ' L * Sqrt(m) / K(m) + End If + angle = Ang + + Else + Msg("error", "Need to specify one more parameter in addition to length") + Return + End If + length = Len + + Else If IsSet("Wid") Then ' if width is specified then... + If IsSet("Ht") Then ' find length & angle based on specified width and height + If Ht < 0 Then + Ht = -Ht ' if height is negative, set it to positive (for the calculations) but flip the reference plane about its x-axis + refPlane.Transform(Transform.Mirror(New Plane(refPlane.Origin, refPlane.XAxis, refPlane.ZAxis))) + flip_A = True + flip_H = True + End If + If Ht = 0 Then ' skip the solver and set the known values + length = Wid + angle = 0 + Else + m = SolveMFromWidHt(Wid, Ht) + length = Cal_L(Ht, m) ' h * K(m) / Sqrt(m) + angle = Cal_A(m) ' Acos(1 - 2 * m) + End If + height = Ht + + Else If IsSet("Ang") Then ' find length & height based on specified width and angle + If Wid = 0 Then + Msg("error", "Curve not possible with width = 0 and an angle as inputs") + Return + End If + If Ang < 0 Then + Ang = -Ang ' if angle is negative, set it to positive (for the calculations) but flip the reference plane about its x-axis + refPlane.Transform(Transform.Mirror(New Plane(refPlane.Origin, refPlane.XAxis, refPlane.ZAxis))) + flip_A = True + flip_H = True + End If + m = Cal_M(Ang) ' (1 - Cos(a)) / 2 + If Ang = 0 Then ' skip the solver and set the known values + length = Wid + height = 0 + Else + length = Wid / (2 * EllipticE(m) / EllipticK(m) - 1) + If length < 0 Then + Msg("error", "Curve not possible at specified width and angle (calculated length is negative)") + Return + End If + height = Cal_H(length, m) ' L * Sqrt(m) / K(m) + End If + angle = Ang + + Else + Msg("error", "Need to specify one more parameter in addition to width (Wid)") + Return + End If + width = Wid + + Else If width IsNot Nothing Then ' if width is determined by PtA and PtB then... + If IsSet("Ht") Then ' find length & angle based on calculated width and height + If Ht < 0 Then + Ht = -Ht ' if height is negative, set it to positive (for the calculations) but flip the reference plane about its x-axis + refPlane.Transform(Transform.Mirror(New Plane(refPlane.Origin, refPlane.XAxis, refPlane.ZAxis))) + flip_A = True + flip_H = True + End If + If Ht = 0 Then ' skip the solver and set the known values + length = width + angle = 0 + Else + m = SolveMFromWidHt(width, Ht) + length = Cal_L(Ht, m) ' h * K(m) / Sqrt(m) + angle = Cal_A(m) ' Acos(1 - 2 * m) + End If + height = Ht + + Else If IsSet("Ang") Then ' find length & height based on calculated width and angle + If width = 0 Then + Msg("error", "Curve not possible with width = 0 and an angle as inputs") + Return + End If + If Ang < 0 Then + Ang = -Ang ' if angle is negative, set it to positive (for the calculations) but flip the reference plane about its x-axis + refPlane.Transform(Transform.Mirror(New Plane(refPlane.Origin, refPlane.XAxis, refPlane.ZAxis))) + flip_A = True + flip_H = True + End If + m = Cal_M(Ang) ' (1 - Cos(a)) / 2 + If Ang = 0 Then ' skip the solver and set the known values + length = width + height = 0 + Else + length = width / (2 * EllipticE(m) / EllipticK(m) - 1) + If length < 0 Then + Msg("error", "Curve not possible at specified width and angle (calculated length is negative)") + Return + End If + height = Cal_H(length, m) ' L * Sqrt(m) / K(m) + End If + angle = Ang + + Else + Msg("error", "Need to specify one more parameter in addition to PtA and PtB") + Return + End If + + Else If IsSet("Ht") Then ' if height is specified then... + If IsSet("Ang") Then ' find length & width based on height and angle + If Ht < 0 Then + Ht = -Ht ' if height is negative, set it to positive (for the calculations) but flip the reference plane about its x-axis + refPlane.Transform(Transform.Mirror(New Plane(refPlane.Origin, refPlane.XAxis, refPlane.ZAxis))) + flip_H = True + flip_A = True + End If + If Ht = 0 Then + Msg("error", "Height can't = 0 if only height and angle are specified") + Return + Else + If Ang < 0 Then + Ang = -Ang ' if angle is negative, set it to positive (for the calculations) but flip the reference plane about its x-axis + refPlane.Transform(Transform.Mirror(New Plane(refPlane.Origin, refPlane.XAxis, refPlane.ZAxis))) + flip_A = Not flip_A + flip_H = Not flip_H + End If + m = Cal_M(Ang) ' (1 - Cos(a)) / 2 + If Ang = 0 Then + Msg("error", "Angle can't = 0 if only height and angle are specified") + Return + Else + length = Cal_L(Ht, m) ' h * K(m) / Sqrt(m) + width = Cal_W(length, m) ' L * (2 * E(m) / K(m) - 1) + End If + angle = Ang + End If + height = Ht + + Else + Msg("error", "Need to specify one more parameter in addition to height") + Return + End If + + Else If IsSet("Ang") Then + Msg("error", "Need to specify one more parameter in addition to angle") + Return + Else + Msg("error", "Need to specify two of the four parameters: length, width (or PtB), height, and angle") + Return + End If + + If m > Defined.M_MAX Then + Msg("error", "Form of curve not solvable with current algorithm and given inputs") + Return + End If + + refPlane.Origin = refPlane.PointAt(width / 2, 0, 0) ' adjust the origin of the reference plane so that the curve is centered about the y-axis (start of the curve is at x = -width/2) + + If multiple_m.Count > 1 Then ' if there is more than one m value returned, calculate the width, angle, and curve for each + Dim multi_pts As New DataTree(Of Point3d) + Dim multi_crv As New List(Of Curve) + Dim tmp_pts As New List(Of Point3d) + Dim multi_W, multi_A, multi_F As New List(Of Double) + Dim j As Integer = 0 ' used for creating a new branch (GH_Path) for storing pts which is itself a list of points + + For Each m_val As Double In multiple_m + width = Cal_W(length, m_val) 'length * (2 * EllipticE(m_val) / EllipticK(m_val) - 1) + + If width < 0 And ignoreSelfIntersecting Then + Msg("warning", "One curve is self-intersecting. To enable these, set ignoreSelfIntersecting to False") + Continue For + End If + + If m_val >= Defined.M_SKETCHY Then Msg("info", "Accuracy of the curve whose width = " & Math.Round(width, 4) & " is not guaranteed") + + angle = Cal_A(m_val) 'Math.Asin(2 * m_val - 1) + refPlane.Origin = refPlane.PointAt(width / 2, 0, 0) ' adjust the origin of the reference plane so that the curve is centered about the y-axis (start of the curve is at x = -width/2) + + tmp_pts = FindBendForm(length, width, m_val, angle, refPlane) + multi_pts.AddRange(tmp_pts, New GH_Path(j)) + multi_crv.Add(MakeCurve(tmp_pts, angle, refPlane)) + + multi_W.Add(width) + If flip_A Then angle = -angle + multi_A.Add(angle) + + E = E * 10 ^ 9 ' Young's modulus input E is in GPa, so we convert to Pa here (= N/m^2) + multi_F.Add(EllipticK(m_val) ^ 2 * E * I / length ^ 2) ' from reference {4} pg. 79 + + j += 1 + refPlane.Origin = PtA ' reset the reference plane origin to PtA for the next m_val + 'Print("length=" & length & ", width=" & width & ", height=" & height & ", angle=" & angle & ", m=" & m_val & ", k=" & Math.Sqrt(m_val) & ", w/L=" & width / length & ", h/L=" & height / length & ", w/h=" & width / height) + Next + + ' assign the outputs + Pts = multi_pts + Crv = multi_crv + L = length + W = multi_W + If flip_H Then height = -height + H = height + A = multi_A + F = multi_F + + Else ' only deal with the single m value + If m >= Defined.M_SKETCHY Then Msg("info", "Accuracy of the curve at these parameters is not guaranteed") + + If width < 0 And ignoreSelfIntersecting Then + Msg("error", "Curve is self-intersecting. To enable these, set ignoreSelfIntersecting to False") + Return + End If + + Pts = FindBendForm(length, width, m, angle, refPlane) + Crv = MakeCurve(pts, angle, refPlane) + L = length + W = width + If flip_H Then height = -height + H = height + If flip_A Then angle = -angle + A = angle + + E = E * 10 ^ 9 ' Young's modulus input E is in GPa, so we convert to Pa here (= N/m^2) + F = EllipticK(m) ^ 2 * E * I / length ^ 2 ' from reference {4} pg. 79. Note: the critical buckling (that makes the rod/wire start to bend) can be found at height=0 (width=length) + + 'height = Math.Sqrt(((2 * Len / 5) ^ 2 - ((Wid - Len / 5) / 2) ^ 2) ' quick approximation discovered by MΓ₯rten of 'Geometry of Bending' fame ( http://tiny.cc/it2pbx ) + 'width = (Len +/- 2 * Math.Sqrt(4 * Len ^ 2 - 25 * Ht ^ 2)) / 5 ' derived from above + 'length = (2 * Math.Sqrt(15 * Ht ^ 2 + 4 * Wid ^ 2) - Wid) / 3 ' derived from above + + 'Print("length=" & length & ", width=" & width & ", height=" & height & ", angle=" & angle & ", m=" & m & ", k=" & Math.Sqrt(m) & ", w/L=" & width / length & ", h/L=" & height / length & ", w/h=" & width / height) + End If + + + + + + + 615 + 822 + 84 + 184 + + + 657 + 914 + + + + + + 9 + 84fa917c-1ed8-4db3-8be1-7bdc4a6495a2 + 84fa917c-1ed8-4db3-8be1-7bdc4a6495a2 + 84fa917c-1ed8-4db3-8be1-7bdc4a6495a2 + 84fa917c-1ed8-4db3-8be1-7bdc4a6495a2 + 84fa917c-1ed8-4db3-8be1-7bdc4a6495a2 + 84fa917c-1ed8-4db3-8be1-7bdc4a6495a2 + 84fa917c-1ed8-4db3-8be1-7bdc4a6495a2 + 84fa917c-1ed8-4db3-8be1-7bdc4a6495a2 + 84fa917c-1ed8-4db3-8be1-7bdc4a6495a2 + 8 + 3ede854e-c753-40eb-84cb-b48008f14fd4 + 8ec86459-bf01-4409-baee-174d0d2b13d0 + 8ec86459-bf01-4409-baee-174d0d2b13d0 + 8ec86459-bf01-4409-baee-174d0d2b13d0 + 8ec86459-bf01-4409-baee-174d0d2b13d0 + 8ec86459-bf01-4409-baee-174d0d2b13d0 + 8ec86459-bf01-4409-baee-174d0d2b13d0 + 8ec86459-bf01-4409-baee-174d0d2b13d0 + + + + + true + Script Variable PtA + e44ef3d7-d80d-4847-b3c9-7145c5d256c7 + PtA + PtA + true + 0 + true + 544607c8-b250-4465-bd1a-e6ed510b2090 + 1 + e1937b56-b1da-4c12-8bd8-e34ee81746ef + + + + + + 617 + 824 + 25 + 20 + + + 631 + 834 + + + + + + + + true + Script Variable PtB + f3a5f81b-cbeb-43a5-b9d7-95ddf8329091 + PtB + PtB + true + 0 + true + 0 + e1937b56-b1da-4c12-8bd8-e34ee81746ef + + + + + + 617 + 844 + 25 + 20 + + + 631 + 854 + + + + + + + + true + Script Variable Pln + 25fe63c8-a9a1-43dd-ba6c-ff444bb1bd38 + Pln + Pln + true + 0 + true + 2e367eec-73c9-49fe-931b-de57feb15198 + 1 + 3897522d-58e9-4d60-b38c-978ddacfedd8 + + + + + + 617 + 864 + 25 + 20 + + + 631 + 874 + + + + + + + + true + Script Variable Len + c645a879-0d63-4de2-8207-39e77ea0b36a + Len + Len + true + 0 + true + 7e001f0c-1ecd-4198-95f7-0d0278815c2b + 1 + 8e991e99-5fb8-41e1-928d-1bba8fb9f7d7 + + + + + + 617 + 884 + 25 + 20 + + + 631 + 894 + + + + + + + + true + Script Variable Wid + 563b27b8-1de0-4076-9540-f1f6287d4585 + Wid + Wid + true + 0 + true + 0 + 8e991e99-5fb8-41e1-928d-1bba8fb9f7d7 + + + + + + 617 + 904 + 25 + 20 + + + 631 + 914 + + + + + + + + true + Script Variable Ht + f945874f-3911-45dd-a65b-5931b59aaeda + Ht + Ht + true + 0 + true + 0 + 8e991e99-5fb8-41e1-928d-1bba8fb9f7d7 + + + + + + 617 + 924 + 25 + 20 + + + 631 + 934 + + + + + + + + true + Script Variable Ang + 7cbb4f70-f9aa-4f07-bc67-aac4eaace9e1 + Ang + Ang + true + 0 + true + b2a8f43d-df9e-4385-bfbe-4e7a54acdc52 + 1 + 8e991e99-5fb8-41e1-928d-1bba8fb9f7d7 + + + + + + 617 + 944 + 25 + 20 + + + 631 + 954 + + + + + + + + true + Script Variable E + 757c9fbf-c87a-438c-9240-7ed119111c17 + E + E + true + 0 + true + 0068d51d-c81e-4187-8df3-5835ab363a73 + 1 + 8e991e99-5fb8-41e1-928d-1bba8fb9f7d7 + + + + + + 617 + 964 + 25 + 20 + + + 631 + 974 + + + + + + + + true + Script Variable I + dfba63c4-6cba-4504-bd15-6d7b03b308e0 + I + I + true + 0 + true + c29e43b1-f147-4596-9a27-65e202efbf44 + 1 + 8e991e99-5fb8-41e1-928d-1bba8fb9f7d7 + + + + + + 617 + 984 + 25 + 20 + + + 631 + 994 + + + + + + + + 1 + Print, Reflect and Error streams + e74891fd-659b-4fdb-833d-cbae9b2064f7 + out + out + false + 0 + + + + + + 672 + 824 + 25 + 22 + + + 684.5 + 835.25 + + + + + + + + Output parameter Pts + 74a1959d-f66e-4c81-8b36-b26ed2379bb0 + Pts + Pts + false + 0 + + + + + + 672 + 846 + 25 + 23 + + + 684.5 + 857.75 + + + + + + + + Output parameter Crv + 9cf1549b-e355-4e34-8c66-56776a6693a3 + Crv + Crv + false + 0 + + + + + + 672 + 869 + 25 + 22 + + + 684.5 + 880.25 + + + + + + + + Output parameter L + 03e53700-d1d7-480a-9312-c1af517baedd + L + L + false + 0 + + + + + + 672 + 891 + 25 + 23 + + + 684.5 + 902.75 + + + + + + + + Output parameter W + 09f794fe-078e-4e7b-840b-9d52b913b097 + W + W + false + 0 + + + + + + 672 + 914 + 25 + 22 + + + 684.5 + 925.25 + + + + + + + + Output parameter H + 2373426a-a151-4361-9f8e-d19fef664dc8 + H + H + false + 0 + + + + + + 672 + 936 + 25 + 23 + + + 684.5 + 947.75 + + + + + + + + Output parameter A + 619887fe-9427-4cea-b7fd-0b8e37ad0966 + A + A + false + 0 + + + + + + 672 + 959 + 25 + 22 + + + 684.5 + 970.25 + + + + + + + + Output parameter F + 20b5374e-e220-4840-8424-dcf1b57d3ad2 + F + F + false + 0 + + + + + + 672 + 981 + 25 + 23 + + + 684.5 + 992.75 + + + + + + + + + + + + + + 57da07bd-ecab-415d-9d86-af36d7073abc + Number Slider + + + + + Numeric slider for single values + 7e001f0c-1ecd-4198-95f7-0d0278815c2b + Number Slider + length + false + 0 + + + + + + 160 + 874 + 382 + 20 + + + 160.4443 + 874.5179 + + + + + + 2 + 1 + 0 + 400 + 0 + 0 + 145.76 + + + + + + + + + fbac3e32-f100-4292-8692-77240a42fd1a + Point + + + + + Contains a collection of three-dimensional points + true + 37000574-d15c-4f7d-92d9-b148cb8b434c + Point + Pt + false + 74a1959d-f66e-4c81-8b36-b26ed2379bb0 + 1 + + + + + + 784 + 791 + 50 + 24 + + + 809.0988 + 803.2372 + + + + + + + + + + 59e0b89a-e487-49f8-bab8-b5bab16be14c + Panel + + + + + A panel for custom notes and text values + 982de226-ec32-4b13-b043-f53881d2e028 + Panel + + false + 0 + 1487d807-5f9b-4abc-9b8d-effa22608b5d + 1 + Double click to edit panel content… + + + + + + 849 + 986 + 105 + 55 + + 0 + 0 + 0 + + 849.9546 + 986.3983 + + + + + + + 255;255;250;90 + + true + true + true + false + false + true + + + + + + + + + 0d77c51e-584f-44e8-aed2-c2ddf4803888 + Degrees + + + + + Convert an angle specified in radians to degrees + f53628df-d187-47f6-8939-9a7cff89ea30 + Degrees + Deg + + + + + + 764 + 997 + 64 + 28 + + + 794 + 1011 + + + + + + Angle in radians + 6e9d9246-ea82-40e7-a5c7-6200a0750140 + Radians + R + false + 619887fe-9427-4cea-b7fd-0b8e37ad0966 + 1 + + + + + + 766 + 999 + 13 + 24 + + + 774 + 1011 + + + + + + + + Angle in degrees + 1487d807-5f9b-4abc-9b8d-effa22608b5d + Degrees + D + false + 0 + + + + + + 809 + 999 + 17 + 24 + + + 817.5 + 1011 + + + + + + + + + + + + 3581f42a-9592-4549-bd6b-1c0fc39d067b + Construct Point + + + + + Construct a point from {xyz} coordinates. + f1c497f2-a656-46bd-97ff-e6f55517ad83 + Construct Point + Pt + + + + + + 390 + 743 + 67 + 64 + + + 421 + 775 + + + + + + {x} coordinate + f274b47b-1218-4782-8247-527101e3221f + X coordinate + X + false + 0 + + + + + + 392 + 745 + 14 + 20 + + + 400.5 + 755 + + + + + + 1 + + + + + 1 + {0} + + + + + 0 + + + + + + + + + + + {y} coordinate + 8bb61e7c-9fbb-42a5-be49-c22980dcde8c + Y coordinate + Y + false + fd7d6e04-d8e1-46ec-9660-5d3b6392bb5c + 1 + + + + + + 392 + 765 + 14 + 20 + + + 400.5 + 775 + + + + + + 1 + + + + + 1 + {0} + + + + + -100 + + + + + + + + + + + {z} coordinate + fd7813a9-1054-41ca-9f6b-83a5fd0efcce + Z coordinate + Z + false + 0 + + + + + + 392 + 785 + 14 + 20 + + + 400.5 + 795 + + + + + + 1 + + + + + 1 + {0} + + + + + 0 + + + + + + + + + + + Point coordinate + 544607c8-b250-4465-bd1a-e6ed510b2090 + Point + Pt + false + 0 + + + + + + 436 + 745 + 19 + 60 + + + 445.5 + 775 + + + + + + + + + + + + d5967b9f-e8ee-436b-a8ad-29fdcecf32d5 + Curve + + + + + Contains a collection of generic curves + 12b4fe3c-7f43-444b-be77-19b89104763b + Curve + Crv + false + 9cf1549b-e355-4e34-8c66-56776a6693a3 + 1 + + + + + + 784 + 825 + 50 + 24 + + + 809.8876 + 837.7669 + + + + + + + + + + 17b7152b-d30d-4d50-b9ef-c9fe25576fc2 + XY Plane + + + + + World XY plane. + true + 8cd68a99-5c0c-40f6-8f6c-c57b65a7c0fb + XY Plane + XY + + + + + + 492 + 837 + 64 + 28 + + + 523 + 851 + + + + + + Origin of plane + d599495e-da0d-4b5f-96ca-d429779f4f7e + Origin + O + false + 544607c8-b250-4465-bd1a-e6ed510b2090 + 1 + + + + + + 494 + 839 + 14 + 24 + + + 502.5 + 851 + + + + + + 1 + + + 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73.81226 + + + -194.0922 + 88.8396 + + + -552.3402 + 88.8396 + + A quick note + Microsoft Sans Serif + 501dc92a-f2db-4bb4-8054-89bd8827333b + false + Scribble + Scribble + 16 + for testing different points on an alternate plane + + + + + + -557.3402 + 68.81226 + 368.248 + 25.02734 + + + -552.3402 + 73.81226 + + + + + + + + + + 7f5c6c55-f846-4a08-9c9a-cfdc285cc6fe + Scribble + + + + + true + + 317.6904 + 47.65552 + + + 768.3664 + 47.65552 + + + 768.3664 + 71.13574 + + + 317.6904 + 71.13574 + + A quick note + Microsoft Sans Serif + 32f0e79e-6dca-4f24-9b30-17ca99e1ac07 + false + Scribble + Scribble + 25 + Elastic Bending Script - Main Example + + + + + + 312.6904 + 42.65552 + 460.676 + 33.48022 + + + 317.6904 + 47.65552 + + + + + + + + + + 7f5c6c55-f846-4a08-9c9a-cfdc285cc6fe + Scribble + + + + + true + + 559.8143 + 683.617 + + + 913.3091 + 683.2137 + + + 913.3262 + 698.241 + + + 559.8313 + 698.6443 + + A quick note + Microsoft Sans Serif + 834f8190-7e8f-4039-8f4f-5e398c86ddfa + false + Scribble + Scribble + 16 + At 60Β°, minimum curve radius = height. Try 90Β° + + + + + + 554.8143 + 678.2137 + 363.5119 + 25.43066 + + + 559.8143 + 683.617 + + + + + + + + + + 079bd9bd-54a0-41d4-98af-db999015f63d + VB Script + + + + + Private Function IsSet(ByVal param As String) As Boolean ' Check if an input parameter has data + Dim i As Integer = Component.Params.IndexOfInputParam(param) + If i > -1 Then + Return Component.Params.Input.ElementAt(i).DataType > 1 ' input parameter DataType of 1 means it's not receiving input (internal or external) + Else + Msg("error", "Input parameter '" & param & "' not found") + Return False + End If + End Function + + Private Sub Msg(ByVal type As String, ByVal msg As String) ' Output an error, warning, or informational message + Select Case type + Case "error" + Component.AddRuntimeMessage(GH_RuntimeMessageLevel.Error, msg) + Print("Error: " & msg) + Case "warning" + Component.AddRuntimeMessage(GH_RuntimeMessageLevel.Warning, msg) + Print("Warning: " & msg) + Case "info" + Component.AddRuntimeMessage(GH_RuntimeMessageLevel.Remark, msg) + Print(msg) + End Select + End Sub + + ' Solve for the m parameter from length and width (reference {1} equation (34), except b = width and K(k) and E(k) should be K(m) and E(m)) + Private Function SolveMFromLenWid(ByVal L As Double, ByVal w As Double) As Double + If w = 0 Then + Return Defined.M_ZERO_W ' for the boundry condition width = 0, bypass the function and return the known m value + End If + + Dim n As Integer = 1 ' Iteration counter (quit if >MAXIT) + Dim lower As Double = 0 ' m must be within this range + Dim upper As Double = 1 + Dim m As Double + Dim cwl As Double + + Do While (upper - lower) > Defined.MAXERR AndAlso (n) < Defined.MAXIT ' Repeat until range narrow enough or MAXIT + m = (upper + lower) / 2 + cwl = 2 * EllipticE(m) / EllipticK(m) - 1 ' calculate w/L with the test value of m + If cwl < w / L Then ' compares the calculated w/L with the actual w/L then narrows the range of possible m + upper = m + Else + lower = m + End If + n += 1 + Loop + Return m + End Function + + ' Solve for the m parameter from length and height (reference {1} equation (33), except K(k) should be K(m) and k = sqrt(m)) + ' Note that it's actually possible to find 2 valid values for m (hence 2 width values) at certain height values + Private Function SolveMFromLenHt(ByVal L As Double, ByVal h As Double) As List(Of Double) + Dim n As Integer = 1 ' Iteration counter (quit if >MAXIT) + Dim lower As Double = 0 ' m must be within this range + Dim upper As Double = 1 + Dim twoWidths As Boolean = h / L >= Defined.DOUBLE_W_HL_RATIO And h / L < Defined.MAX_HL_RATIO ' check to see if h/L is within the range where 2 solutions for the width are possible + Dim m As Double + Dim mult_m As New List(Of Double) + Dim chl As Double + + If twoWidths Then + ' find the first of two possible solutions for m with the following limits: + lower = Defined.M_DOUBLE_W ' see constants at bottom of script + upper = Defined.M_MAXHEIGHT ' see constants at bottom of script + Do While (upper - lower) > Defined.MAXERR AndAlso (n) < Defined.MAXIT ' Repeat until range narrow enough or MAXIT + m = (upper + lower) / 2 + chl = Math.Sqrt(m) / EllipticK(m) ' calculate h/L with the test value of m + If chl > h / L Then ' compares the calculated h/L with the actual h/L then narrows the range of possible m + upper = m + Else + lower = m + End If + n += 1 + Loop + mult_m.Add(m) + + ' then find the second of two possible solutions for m with the following limits: + lower = Defined.M_MAXHEIGHT ' see constants at bottom of script + upper = 1 + Do While (upper - lower) > Defined.MAXERR AndAlso (n) < Defined.MAXIT ' Repeat until range narrow enough or MAXIT + m = (upper + lower) / 2 + chl = Math.Sqrt(m) / EllipticK(m) ' calculate h/L with the test value of m + If chl < h / L Then ' compares the calculated h/L with the actual h/L then narrows the range of possible m + upper = m + Else + lower = m + End If + n += 1 + Loop + + If m <= Defined.M_MAX Then ' return this m parameter only if it falls within the maximum useful value (above which the curve breaks down) + mult_m.Add(m) + End If + + Else + ' find the one possible solution for the m parameter + upper = Defined.M_DOUBLE_W ' limit the upper end of the search to the maximum value of m for which only one solution exists + Do While (upper - lower) > Defined.MAXERR AndAlso (n) < Defined.MAXIT ' Repeat until range narrow enough or MAXIT + m = (upper + lower) / 2 + chl = Math.Sqrt(m) / EllipticK(m) ' calculate h/L with the test value of m + If chl > h / L Then ' compares the calculated h/L with the actual h/L then narrows the range of possible m + upper = m + Else + lower = m + End If + n += 1 + Loop + mult_m.Add(m) + End If + + Return mult_m + End Function + + ' Solve for the m parameter from width and height (derived from reference {1} equations (33) and (34) with same notes as above) + Private Function SolveMFromWidHt(ByVal w As Double, ByVal h As Double) As Double + Dim n As Integer = 1 ' Iteration counter (quit if >MAXIT) + Dim lower As Double = 0 ' m must be within this range + Dim upper As Double = 1 + Dim m As Double + Dim cwh As Double + + Do While (upper - lower) > Defined.MAXERR AndAlso (n) < Defined.MAXIT ' Repeat until range narrow enough or MAXIT + m = (upper + lower) / 2 + cwh = (2 * EllipticE(m) - EllipticK(m)) / Math.Sqrt(m) ' calculate w/h with the test value of m + If cwh < w / h Then ' compares the calculated w/h with the actual w/h then narrows the range of possible m + upper = m + Else + lower = m + End If + n += 1 + Loop + + Return m + End Function + + ' Calculate length based on height and an m parameter, derived from reference {1} equation (33), except K(k) should be K(m) and k = sqrt(m) + Private Function Cal_L(ByVal h As Double, ByVal m As Double) As Double + Return h * EllipticK(m) / Math.Sqrt(m) + End Function + + ' Calculate width based on length and an m parameter, derived from reference {1} equation (34), except b = width and K(k) and E(k) should be K(m) and E(m) + Private Function Cal_W(ByVal L As Double, ByVal m As Double) As Double + Return L * (2 * EllipticE(m) / EllipticK(m) - 1) + End Function + + ' Calculate height based on length and an m parameter, from reference {1} equation (33), except K(k) should be K(m) and k = sqrt(m) + Private Function Cal_H(ByVal L As Double, ByVal m As Double) As Double + Return L * Math.Sqrt(m) / EllipticK(m) + End Function + + ' Calculate the unique m parameter based on a start tangent angle, from reference {2}, just above equation (9a), that states k = Sin(angle / 2 + Pi / 4), + ' but as m = k^2 and due to this script's need for an angle rotated 90Β° versus the one in reference {1}, the following formula is the result + ' New note: verified by reference {4}, pg. 78 at the bottom + Private Function Cal_M(ByVal a As Double) As Double + Return (1 - Math.Cos(a)) / 2 ' equal to Sin^2(a/2) too + End Function + + ' Calculate start tangent angle based on an m parameter, derived from above formula + Private Function Cal_A(ByVal m As Double) As Double + Return Math.Acos(1 - 2 * m) + End Function + + ' This is the heart of this script, taking the found (or specified) length, width, and angle values along with the found m parameter to create + ' a list of points that approximate the shape or form of the elastica. It works by finding the x and y coordinates (which are reversed versus + ' the original equations (12a) and (12b) from reference {2} due to the 90Β° difference in orientation) based on the tangent angle along the curve. + ' See reference {2} for more details on how they derived it. Note that to simplify things, the algorithm only calculates the points for half of the + ' curve, then mirrors those points along the y-axis. + Private Function FindBendForm(ByVal L As Double, ByVal w As Double, ByVal m As Double, ByVal ang As Double, ByVal refPln As Plane) As List(Of Point3d) + L = L / 2 ' because the below algorithm is based on the formulas in reference {2} for only half of the curve + w = w / 2 ' same + + If ang = 0 Then ' if angle (and height) = 0, then simply return the start and end points of the straight line + Dim out As New List(Of Point3d) + out.Add(refPln.PointAt(w, 0, 0)) + out.Add(refPln.PointAt(-w, 0, 0)) + Return out + End If + + Dim x As Double + Dim y As Double + Dim halfCurvePts As New List(Of Point3d) + Dim fullCurvePts As New List(Of Point3d) + Dim translatedPts As New List(Of Point3d) + + ang -= Math.PI / 2 ' a hack to allow this algorithm to work, since the original curve in paper {2} was rotated 90Β° + Dim angB As Double = ang + (-Math.PI / 2 - ang) / Defined.CURVEDIVS ' angB is the 'lowercase theta' which should be in formula {2}(12b) as the interval + ' start [a typo...see equation(3)]. It's necessary to start angB at ang + [interval] instead of just ang due to integration failing at angB = ang + halfCurvePts.Add(New Point3d(w, 0, 0)) ' start with this known initial point, as integration will fail when angB = ang + + ' each point {x, y} is calculated from the tangent angle, angB, that occurs at each point (which is why this iterates from ~ang to -pi/2, the known end condition) + Do While Math.Round(angB, Defined.ROUNDTO) >= Math.Round(-Math.PI / 2, Defined.ROUNDTO) + y = (Math.Sqrt(2) * Math.Sqrt(Math.Sin(ang) - Math.Sin(angB)) * (w + L)) / (2 * EllipticE(m)) ' note that x and y are swapped vs. (12a) and (12b) + x = (L / (Math.Sqrt(2) * EllipticK(m))) * Simpson(angB, -Math.PI / 2, 500, ang) ' calculate the Simpson approximation of the integral (function f below) + ' over the interval angB ('lowercase theta') to -pi/2. side note: is 500 too few iterations for the Simson algorithm? + + If Math.Round(x, Defined.ROUNDTO) = 0 Then x = 0 + halfCurvePts.Add(New Point3d(x, y, 0)) + + angB += (-Math.PI / 2 - ang) / Defined.CURVEDIVS ' onto the next tangent angle + Loop + + ' After finding the x and y values for half of the curve, add the {-x, y} values for the rest of the curve + For Each point As Point3d In halfCurvePts + If Math.Round(point.X, Defined.ROUNDTO) = 0 Then + If Math.Round(point.Y, Defined.ROUNDTO) = 0 Then + fullCurvePts.Add(New Point3d(0, 0, 0)) ' special case when width = 0: when x = 0, only duplicate the point when y = 0 too + End If + Else + fullCurvePts.Add(New Point3d(-point.X, point.Y, 0)) + End If + Next + halfCurvePts.Reverse + fullCurvePts.AddRange(halfCurvePts) + + For Each p As Point3d In fullCurvePts + translatedPts.Add(refPln.PointAt(p.X, p.Y, p.Z)) ' translate the points from the reference plane to the world plane + Next + + Return translatedPts + End Function + + ' Interpolates the points from FindBendForm to create the Elastica curve. Uses start & end tangents for greater accuracy. + Private Function MakeCurve(ByVal pts As List(Of Point3d), ByVal ang As Double, ByVal refPln As Plane) As Curve + If ang <> 0 Then + Dim ts, te As New Vector3d(refPln.XAxis) + ts.Rotate(ang, refPln.ZAxis) + te.Rotate(-ang, refPln.ZAxis) + Return Curve.CreateInterpolatedCurve(pts, 3, CurveKnotStyle.Chord, ts, te) ' 3rd degree curve with 'Chord' Knot Style + Else + Return Curve.CreateInterpolatedCurve(pts, 3) ' if angle (and height) = 0, then simply interpolate the straight line (no start/end tangents) + End If + End Function + + ' Implements the Simpson approximation for an integral of function f below + Public Function Simpson(a As Double, b As Double, n As Integer, theta As Double) As Double 'n should be an even number + Dim j As Integer, s1 As Double, s2 As Double, h As Double + h = (b - a) / n + s1 = 0 + s2 = 0 + For j = 1 To n - 1 Step 2 + s1 = s1 + fn(a + j * h, theta) + Next j + For j = 2 To n - 2 Step 2 + s2 = s2 + fn(a + j * h, theta) + Next j + Simpson = h / 3 * (fn(a, theta) + 4 * s1 + 2 * s2 + fn(b, theta)) + End Function + + ' Specific calculation for the above integration + Public Function fn(x As Double, theta As Double) As Double + fn = Math.Sin(x) / (Math.Sqrt(Math.Sin(theta) - Math.Sin(x))) ' from reference {2} formula (12b) + End Function + + + ' Return the Complete Elliptic integral of the 1st kind + ' Abramowitz and Stegun p.591, formula 17.3.11 + ' Code from http://www.codeproject.com/Articles/566614/Elliptic-integrals + Public Function EllipticK(ByVal m As Double) As Double + Dim sum, term, above, below As Double + sum = 1 + term = 1 + above = 1 + below = 2 + + For i As Integer = 1 To 100 + term *= above / below + sum += Math.Pow(m, i) * Math.Pow(term, 2) + above += 2 + below += 2 + Next + sum *= 0.5 * Math.PI + Return sum + End Function + + + ' Return the Complete Elliptic integral of the 2nd kind + ' Abramowitz and Stegun p.591, formula 17.3.12 + ' Code from http://www.codeproject.com/Articles/566614/Elliptic-integrals + Public Function EllipticE(ByVal m As Double) As Double + Dim sum, term, above, below As Double + sum = 1 + term = 1 + above = 1 + below = 2 + + For i As Integer = 1 To 100 + term *= above / below + sum -= Math.Pow(m, i) * Math.Pow(term, 2) / above + above += 2 + below += 2 + Next + sum *= 0.5 * Math.PI + Return sum + End Function + + Friend Partial NotInheritable Class Defined + Private Sub New() + End Sub + + ' Note: most of these values for m and h/L ratio were found with Wolfram Alpha and either specific intercepts (x=0) or local minima/maxima. They should be constant. + Public Const M_SKETCHY As Double = 0.95 ' value of the m parameter where the curvature near the ends of the curve gets wonky + Public Const M_MAX As Double = 0.993 ' maximum useful value of the m parameter, above which this algorithm for the form of the curve breaks down + Public Const M_ZERO_W As Double = 0.826114765984970336 ' value of the m parameter when width = 0 + Public Const M_MAXHEIGHT As Double = 0.701327460663101223 ' value of the m parameter at maximum possible height of the bent rod/wire + Public Const M_DOUBLE_W As Double = 0.180254422335013983 ' minimum value of the m parameter when two width values are possible for a given height and length + Public Const DOUBLE_W_HL_RATIO As Double = 0.257342117984635757 ' value of the height/length ratio above which there are two possible width values + Public Const MAX_HL_RATIO As Double = 0.403140189705650243 ' maximum possible value of the height/length ratio + + Public Const MAXERR As Double = 0.0000000001 ' error tolerance + Public Const MAXIT As Integer = 100 ' maximum number of iterations + Public Const ROUNDTO As Integer = 10 ' number of decimal places to round off to + Public Const CURVEDIVS As Integer = 50 ' number of sample points for building the curve (or half-curve as it were) + End Class + A VB.NET scriptable component + + 98 + 86 + + true + bf1f4616-5fd9-426e-9474-52a076d17bf4 + VB Script + VB + true + 0 + ' ----------------------------------------------------------------- + ' Elastic Bending Script by Will McElwain + ' Created February 2014 + ' + ' DESCRIPTION: + ' This beast creates the so-called 'elastica curve', the shape a long, thin rod or wire makes when it is bent elastically (i.e. not permanently). In this case, force + ' is assumed to only be applied horizontally (which would be in line with the rod at rest) and both ends are assumed to be pinned or hinged meaning they are free + ' to rotate (as opposed to clamped, when the end tangent angle is fixed, usually horizontally). An interesting finding is that it doesn't matter what the material or + ' cross-sectional area is, as long as they're uniform along the entire length. Everything makes the same shape when bent as long as it doesn't cross the threshold + ' from elastic to plastic (permanent) deformation (I don't bother to find that limit here, but can be found if the yield stress for a material is known). + ' + ' Key to the formulas used in this script are elliptic integrals, specifically K(m), the complete elliptic integral of the first kind, and E(m), the complete elliptic + ' integral of the second kind. There was a lot of confusion over the 'm' and 'k' parameters for these functions, as some people use them interchangeably, but they are + ' not the same. m = k^2 (thus k = Sqrt(m)). I try to use the 'm' parameter exclusively to avoid this confusion. Note that there is a unique 'm' parameter for every + ' configuration/shape of the elastica curve. + ' + ' This script tries to find that unique 'm' parameter based on the inputs. The algorithm starts with a test version of m, evaluates an expression, say 2*E(m)/K(m)-1, + ' then compares the result to what it should be (in this case, a known width/length ratio). Iterate until the correct m is found. Once we have m, we can then calculate + ' all of the other unknowns, then find points that lie on that curve, then interpolate those points for the actual curve. You can also use Wolfram|Alpha as I did to + ' find the m parameter based on the equations in this script (example here: http://tiny.cc/t4tpbx for when say width=45.2 and length=67.1). + ' + ' Other notes: + ' * This script works with negative values for width, which will creat a self-intersecting curve (as it should). The curvature of the elastica starts to break down around + ' m=0.95 (~154Β°), but this script will continue to work until M_MAX, m=0.993 (~169Β°). If you wish to ignore self-intersecting curves, set ignoreSelfIntersecting to True + ' * When the only known values are length and height, it is actually possible for certain ratios of height to length to have two valid m values (thus 2 possible widths + ' and angles). This script will return them both. + ' * Only the first two valid parameters (of the required ones) will be used, meaning if all four are connected (length, width or a PtB, height, and angle), this script will + ' only use length and width (or a PtB). + ' * Depending on the magnitude of your inputs (say if they're really small, like if length < 10), you might have to increase the constant ROUNDTO at the bottom + ' + ' REFERENCES: + ' {1} "The elastic rod" by M.E. Pacheco Q. & E. Pina, http://www.scielo.org.mx/pdf/rmfe/v53n2/v53n2a8.pdf + ' {2} "An experiment in nonlinear beam theory" by A. Valiente, http://www.deepdyve.com/lp/doc/I3lwnxdfGz , also here: http://tiny.cc/Valiente_AEiNBT + ' {3} "Snap buckling, writhing and Loop formation In twisted rods" by V.G.A. GOSS, http://myweb.lsbu.ac.uk/~gossga/thesisFinal.pdf + ' {4} "Theory of Elastic Stability" by Stephen Timoshenko, http://www.scribd.com/doc/50402462/Timoshenko-Theory-of-Elastic-Stability (start on p. 76) + ' + ' INPUT: + ' PtA - First anchor point (required) + ' PtB - Second anchor point (optional, though 2 out of the 4--length, width, height, angle--need to be specified) + ' [note that PtB can be the same as PtA (meaning width would be zero)] + ' [also note that if a different width is additionally specified that's not equal to the distance between PtA and PtB, then the end point will not equal PtB anymore] + ' Pln - Plane of the bent rod/wire, which bends up in the +y direction. The line between PtA and PtB (if specified) must be parallel to the x-axis of this plane + ' + ' ** 2 of the following 4 need to be specified ** + ' Len - Length of the rod/wire, which needs to be > 0 + ' Wid - Width between the endpoints of the curve [note: if PtB is specified in addition, and distance between PtA and PtB <> width, the end point will be relocated + ' Ht - Height of the bent rod/wire (when negative, curve will bend downward, relative to the input plane, instead) + ' Ang - Inner departure angle or tangent angle (in radians) at the ends of the bent rod/wire. Set up so as width approaches length (thus height approaches zero), angle approaches zero + ' + ' * Following variables only needed for optional calculating of bending force, not for shape of curve. + ' E - Young's modulus (modulus of elasticity) in GPa (=N/m^2) (material-specific. for example, 7075 aluminum is roughly 71.7 GPa) + ' I - Second moment of area (or area moment of inertia) in m^4 (cross-section-specific. for example, a hollow rod + ' would have I = pi * (outer_diameter^4 - inner_diameter^4) / 32 + ' Note: E*I is also known as flexural rigidity or bending stiffness + ' + ' OUTPUT: + ' out - only for debugging messages + ' Pts - the list of points that approximate the shape of the elastica + ' Crv - the 3rd-degree curve interpolated from those points (with accurate start & end tangents) + ' L - the length of the rod/wire + ' W - the distance (width) between the endpoints of the rod/wire + ' H - the height of the bent rod/wire + ' A - the tangent angle at the (start) end of the rod/wire + ' F - the force needed to hold the rod/wire in a specific shape (based on the material properties & cross-section) **be sure your units for 'I' match your units for the + ' rest of your inputs (length, width, etc.). Also note that the critical buckling load (force) that makes the rod/wire start to bend can be found at height=0 + ' + ' THANKS TO: + ' MΓ₯rten Nettelbladt (thegeometryofbending.blogspot.com) + ' Daniel Piker (Kangaroo plugin) + ' David Rutten (Grasshopper guru) + ' Euler & Bernoulli (the O.G.'s) + ' + ' ----------------------------------------------------------------- + + Dim ignoreSelfIntersecting As Boolean = False ' set to True if you don't want to output curves where width < 0, which creates a self-intersecting curve + + Dim inCt As Integer = 0 ' count the number of required parameters that are receiving data + Dim length As Double + Dim width As System.Object = Nothing ' need to set as Nothing so we can check if it has been assigned a value later + Dim height As Double + Dim angle As Double + Dim m As Double + Dim multiple_m As New List(Of Double) + Dim AtoB As Line + Dim flip_H As Boolean = False ' if height is negative, this flag will be set + Dim flip_A As Boolean = False ' if angle is negative, this flag will be set + + If Not IsSet("Pln") Then + Msg("error", "Base plane is not set") + Return + End If + + If Not IsSet("PtA") Then + Msg("error", "Point A is not set") + Return + End If + + If Math.Round(Pln.DistanceTo(PtA), Defined.ROUNDTO) <> 0 Then + Msg("error", "Point A is not on the base plane") + Return + End If + + Dim refPlane As Plane = Pln ' create a reference plane = input plane and set the origin of it to PtA in case PtA isn't the origin already + refPlane.Origin = PtA + + If IsSet("PtB") Then + If Math.Round(Pln.DistanceTo(PtB), Defined.ROUNDTO) <> 0 Then + Msg("error", "Point B is not on the base plane") + Return + End If + + AtoB = New Line(PtA, PtB) + If AtoB.Length <> 0 And Not AtoB.Direction.IsPerpendicularTo(Pln.YAxis) Then + Msg("error", "The line between PtA and PtB is not perpendicular to the Y-axis of the specified plane") + Return + End If + + inCt += 1 + If IsSet("Wid") Then Msg("info", "Wid will override the distance between PtA and PtB. If you do not want this to happen, disconnect PtB or Wid.") + + width = PtA.DistanceTo(PtB) ' get the width (distance) between PtA and PtB + + Dim refPtB As Point3d + refPlane.RemapToPlaneSpace(PtB, refPtB) + If refPtB.X < 0 Then width = -width ' check if PtB is to the left of PtA...if so, width is negative + End If + + If IsSet("Len") Then inCt += 1 + If IsSet("Wid") Then inCt += 1 + If IsSet("Ht") Then inCt += 1 + If IsSet("Ang") Then inCt += 1 + If inCt > 2 Then Msg("info", "More parameters set than are required (out of length, width, height, angle). Only using the first two valid ones.") + + ' check for connected/specified inputs. note: only the first two that it comes across will be used + If IsSet("Len") Then ' if length is specified then... + If Len <= 0 Then + Msg("error", "Length cannot be negative or zero") + Return + End If + If IsSet("Wid") Then ' find height & angle based on length and specified width + If Wid > Len Then + Msg("error", "Width is greater than length") + Return + End If + If Wid = Len Then ' skip the solver and set the known values + height = 0 + m = 0 + angle = 0 + width = Wid + Else + m = SolveMFromLenWid(Len, Wid) + height = Cal_H(Len, m) ' L * Sqrt(m) / K(m) + angle = Cal_A(m) ' Acos(1 - 2 * m) + width = Wid + End If + + Else If width IsNot Nothing Then ' find height & angle based on length and calculated width (distance between PtA and PtB) + If width > Len Then + Msg("error", "Width is greater than length") + Return + End If + If width = Len Then ' skip the solver and set the known values + height = 0 + m = 0 + angle = 0 + Else + m = SolveMFromLenWid(Len, width) + height = Cal_H(Len, m) ' L * Sqrt(m) / K(m) + angle = Cal_A(m) ' Acos(1 - 2 * m) + End If + + Else If IsSet("Ht") Then ' find width & angle based on length and height ** possible to return 2 results ** + If Math.Abs(Ht / Len) > Defined.MAX_HL_RATIO Then + Msg("error", "Height not possible with given length") + Return + End If + If Ht < 0 Then + Ht = -Ht ' if height is negative, set it to positive (for the calculations) but flip the reference plane about its x-axis + refPlane.Transform(Transform.Mirror(New Plane(refPlane.Origin, refPlane.XAxis, refPlane.ZAxis))) + flip_A = True + flip_H = True + End If + If Ht = 0 Then ' skip the solver and set the known values + width = Len + angle = 0 + Else + multiple_m = SolveMFromLenHt(Len, Ht) ' note that it's possible for two values of m to be found if height is close to max height + If multiple_m.Count = 1 Then ' if there's only one m value returned, calculate the width & angle here. we'll deal with multiple m values later + m = multiple_m.Item(0) + width = Cal_W(Len, m) ' L * (2 * E(m) / K(m) - 1) + angle = Cal_A(m) ' Acos(1 - 2 * m) + End If + End If + height = Ht + + Else If IsSet("Ang") Then ' find width & height based on length and angle + If Ang < 0 Then + Ang = -Ang ' if angle is negative, set it to positive (for the calculations) but flip the reference plane about its x-axis + refPlane.Transform(Transform.Mirror(New Plane(refPlane.Origin, refPlane.XAxis, refPlane.ZAxis))) + flip_A = True + flip_H = True + End If + m = Cal_M(Ang) ' (1 - Cos(a)) / 2 + If Ang = 0 Then ' skip the solver and set the known values + width = Len + height = 0 + Else + width = Cal_W(Len, m) ' L * (2 * E(m) / K(m) - 1) + height = Cal_H(Len, m) ' L * Sqrt(m) / K(m) + End If + angle = Ang + + Else + Msg("error", "Need to specify one more parameter in addition to length") + Return + End If + length = Len + + Else If IsSet("Wid") Then ' if width is specified then... + If IsSet("Ht") Then ' find length & angle based on specified width and height + If Ht < 0 Then + Ht = -Ht ' if height is negative, set it to positive (for the calculations) but flip the reference plane about its x-axis + refPlane.Transform(Transform.Mirror(New Plane(refPlane.Origin, refPlane.XAxis, refPlane.ZAxis))) + flip_A = True + flip_H = True + End If + If Ht = 0 Then ' skip the solver and set the known values + length = Wid + angle = 0 + Else + m = SolveMFromWidHt(Wid, Ht) + length = Cal_L(Ht, m) ' h * K(m) / Sqrt(m) + angle = Cal_A(m) ' Acos(1 - 2 * m) + End If + height = Ht + + Else If IsSet("Ang") Then ' find length & height based on specified width and angle + If Wid = 0 Then + Msg("error", "Curve not possible with width = 0 and an angle as inputs") + Return + End If + If Ang < 0 Then + Ang = -Ang ' if angle is negative, set it to positive (for the calculations) but flip the reference plane about its x-axis + refPlane.Transform(Transform.Mirror(New Plane(refPlane.Origin, refPlane.XAxis, refPlane.ZAxis))) + flip_A = True + flip_H = True + End If + m = Cal_M(Ang) ' (1 - Cos(a)) / 2 + If Ang = 0 Then ' skip the solver and set the known values + length = Wid + height = 0 + Else + length = Wid / (2 * EllipticE(m) / EllipticK(m) - 1) + If length < 0 Then + Msg("error", "Curve not possible at specified width and angle (calculated length is negative)") + Return + End If + height = Cal_H(length, m) ' L * Sqrt(m) / K(m) + End If + angle = Ang + + Else + Msg("error", "Need to specify one more parameter in addition to width (Wid)") + Return + End If + width = Wid + + Else If width IsNot Nothing Then ' if width is determined by PtA and PtB then... + If IsSet("Ht") Then ' find length & angle based on calculated width and height + If Ht < 0 Then + Ht = -Ht ' if height is negative, set it to positive (for the calculations) but flip the reference plane about its x-axis + refPlane.Transform(Transform.Mirror(New Plane(refPlane.Origin, refPlane.XAxis, refPlane.ZAxis))) + flip_A = True + flip_H = True + End If + If Ht = 0 Then ' skip the solver and set the known values + length = width + angle = 0 + Else + m = SolveMFromWidHt(width, Ht) + length = Cal_L(Ht, m) ' h * K(m) / Sqrt(m) + angle = Cal_A(m) ' Acos(1 - 2 * m) + End If + height = Ht + + Else If IsSet("Ang") Then ' find length & height based on calculated width and angle + If width = 0 Then + Msg("error", "Curve not possible with width = 0 and an angle as inputs") + Return + End If + If Ang < 0 Then + Ang = -Ang ' if angle is negative, set it to positive (for the calculations) but flip the reference plane about its x-axis + refPlane.Transform(Transform.Mirror(New Plane(refPlane.Origin, refPlane.XAxis, refPlane.ZAxis))) + flip_A = True + flip_H = True + End If + m = Cal_M(Ang) ' (1 - Cos(a)) / 2 + If Ang = 0 Then ' skip the solver and set the known values + length = width + height = 0 + Else + length = width / (2 * EllipticE(m) / EllipticK(m) - 1) + If length < 0 Then + Msg("error", "Curve not possible at specified width and angle (calculated length is negative)") + Return + End If + height = Cal_H(length, m) ' L * Sqrt(m) / K(m) + End If + angle = Ang + + Else + Msg("error", "Need to specify one more parameter in addition to PtA and PtB") + Return + End If + + Else If IsSet("Ht") Then ' if height is specified then... + If IsSet("Ang") Then ' find length & width based on height and angle + If Ht < 0 Then + Ht = -Ht ' if height is negative, set it to positive (for the calculations) but flip the reference plane about its x-axis + refPlane.Transform(Transform.Mirror(New Plane(refPlane.Origin, refPlane.XAxis, refPlane.ZAxis))) + flip_H = True + flip_A = True + End If + If Ht = 0 Then + Msg("error", "Height can't = 0 if only height and angle are specified") + Return + Else + If Ang < 0 Then + Ang = -Ang ' if angle is negative, set it to positive (for the calculations) but flip the reference plane about its x-axis + refPlane.Transform(Transform.Mirror(New Plane(refPlane.Origin, refPlane.XAxis, refPlane.ZAxis))) + flip_A = Not flip_A + flip_H = Not flip_H + End If + m = Cal_M(Ang) ' (1 - Cos(a)) / 2 + If Ang = 0 Then + Msg("error", "Angle can't = 0 if only height and angle are specified") + Return + Else + length = Cal_L(Ht, m) ' h * K(m) / Sqrt(m) + width = Cal_W(length, m) ' L * (2 * E(m) / K(m) - 1) + End If + angle = Ang + End If + height = Ht + + Else + Msg("error", "Need to specify one more parameter in addition to height") + Return + End If + + Else If IsSet("Ang") Then + Msg("error", "Need to specify one more parameter in addition to angle") + Return + Else + Msg("error", "Need to specify two of the four parameters: length, width (or PtB), height, and angle") + Return + End If + + If m > Defined.M_MAX Then + Msg("error", "Form of curve not solvable with current algorithm and given inputs") + Return + End If + + refPlane.Origin = refPlane.PointAt(width / 2, 0, 0) ' adjust the origin of the reference plane so that the curve is centered about the y-axis (start of the curve is at x = -width/2) + + If multiple_m.Count > 1 Then ' if there is more than one m value returned, calculate the width, angle, and curve for each + Dim multi_pts As New DataTree(Of Point3d) + Dim multi_crv As New List(Of Curve) + Dim tmp_pts As New List(Of Point3d) + Dim multi_W, multi_A, multi_F As New List(Of Double) + Dim j As Integer = 0 ' used for creating a new branch (GH_Path) for storing pts which is itself a list of points + + For Each m_val As Double In multiple_m + width = Cal_W(length, m_val) 'length * (2 * EllipticE(m_val) / EllipticK(m_val) - 1) + + If width < 0 And ignoreSelfIntersecting Then + Msg("warning", "One curve is self-intersecting. To enable these, set ignoreSelfIntersecting to False") + Continue For + End If + + If m_val >= Defined.M_SKETCHY Then Msg("info", "Accuracy of the curve whose width = " & Math.Round(width, 4) & " is not guaranteed") + + angle = Cal_A(m_val) 'Math.Asin(2 * m_val - 1) + refPlane.Origin = refPlane.PointAt(width / 2, 0, 0) ' adjust the origin of the reference plane so that the curve is centered about the y-axis (start of the curve is at x = -width/2) + + tmp_pts = FindBendForm(length, width, m_val, angle, refPlane) + multi_pts.AddRange(tmp_pts, New GH_Path(j)) + multi_crv.Add(MakeCurve(tmp_pts, angle, refPlane)) + + multi_W.Add(width) + If flip_A Then angle = -angle + multi_A.Add(angle) + + E = E * 10 ^ 9 ' Young's modulus input E is in GPa, so we convert to Pa here (= N/m^2) + multi_F.Add(EllipticK(m_val) ^ 2 * E * I / length ^ 2) ' from reference {4} pg. 79 + + j += 1 + refPlane.Origin = PtA ' reset the reference plane origin to PtA for the next m_val + 'Print("length=" & length & ", width=" & width & ", height=" & height & ", angle=" & angle & ", m=" & m_val & ", k=" & Math.Sqrt(m_val) & ", w/L=" & width / length & ", h/L=" & height / length & ", w/h=" & width / height) + Next + + ' assign the outputs + Pts = multi_pts + Crv = multi_crv + L = length + W = multi_W + If flip_H Then height = -height + H = height + A = multi_A + F = multi_F + + Else ' only deal with the single m value + If m >= Defined.M_SKETCHY Then Msg("info", "Accuracy of the curve at these parameters is not guaranteed") + + If width < 0 And ignoreSelfIntersecting Then + Msg("error", "Curve is self-intersecting. To enable these, set ignoreSelfIntersecting to False") + Return + End If + + Pts = FindBendForm(length, width, m, angle, refPlane) + Crv = MakeCurve(pts, angle, refPlane) + L = length + W = width + If flip_H Then height = -height + H = height + If flip_A Then angle = -angle + A = angle + + E = E * 10 ^ 9 ' Young's modulus input E is in GPa, so we convert to Pa here (= N/m^2) + F = EllipticK(m) ^ 2 * E * I / length ^ 2 ' from reference {4} pg. 79. Note: the critical buckling (that makes the rod/wire start to bend) can be found at height=0 (width=length) + + 'height = Math.Sqrt(((2 * Len / 5) ^ 2 - ((Wid - Len / 5) / 2) ^ 2) ' quick approximation discovered by MΓ₯rten of 'Geometry of Bending' fame ( http://tiny.cc/it2pbx ) + 'width = (Len +/- 2 * Math.Sqrt(4 * Len ^ 2 - 25 * Ht ^ 2)) / 5 ' derived from above + 'length = (2 * Math.Sqrt(15 * Ht ^ 2 + 4 * Wid ^ 2) - Wid) / 3 ' derived from above + + 'Print("length=" & length & ", width=" & width & ", height=" & height & ", angle=" & angle & ", m=" & m & ", k=" & Math.Sqrt(m) & ", w/L=" & width / length & ", h/L=" & height / length & ", w/h=" & width / height) + End If + + + + + + + 618 + 1502 + 84 + 184 + + + 660 + 1594 + + + + + + 9 + 84fa917c-1ed8-4db3-8be1-7bdc4a6495a2 + 84fa917c-1ed8-4db3-8be1-7bdc4a6495a2 + 84fa917c-1ed8-4db3-8be1-7bdc4a6495a2 + 84fa917c-1ed8-4db3-8be1-7bdc4a6495a2 + 84fa917c-1ed8-4db3-8be1-7bdc4a6495a2 + 84fa917c-1ed8-4db3-8be1-7bdc4a6495a2 + 84fa917c-1ed8-4db3-8be1-7bdc4a6495a2 + 84fa917c-1ed8-4db3-8be1-7bdc4a6495a2 + 84fa917c-1ed8-4db3-8be1-7bdc4a6495a2 + 8 + 3ede854e-c753-40eb-84cb-b48008f14fd4 + 8ec86459-bf01-4409-baee-174d0d2b13d0 + 8ec86459-bf01-4409-baee-174d0d2b13d0 + 8ec86459-bf01-4409-baee-174d0d2b13d0 + 8ec86459-bf01-4409-baee-174d0d2b13d0 + 8ec86459-bf01-4409-baee-174d0d2b13d0 + 8ec86459-bf01-4409-baee-174d0d2b13d0 + 8ec86459-bf01-4409-baee-174d0d2b13d0 + + + + + true + Script Variable PtA + fc00bec5-e331-4012-b0a8-a6f9d0f686f7 + PtA + PtA + true + 0 + true + 7470aaae-fe5c-4a6e-a5d7-3c8c950bb9fb + 1 + e1937b56-b1da-4c12-8bd8-e34ee81746ef + + + + + + 620 + 1504 + 25 + 20 + + + 634 + 1514 + + + + + + + + true + Script Variable PtB + c5bc96c5-9e28-4cb8-9259-356c9db2b9fb + PtB + PtB + true + 0 + true + 0 + e1937b56-b1da-4c12-8bd8-e34ee81746ef + + + + + + 620 + 1524 + 25 + 20 + + + 634 + 1534 + + + + + + + + true + Script Variable Pln + 71f32d80-b186-4e03-b761-b9bb960ea743 + Pln + Pln + true + 0 + true + 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+ + + + + 675 + 1571 + 25 + 23 + + + 687.5 + 1582.75 + + + + + + + + Output parameter W + 0c900729-ed6e-40e8-809f-e2432521ac54 + W + W + false + 0 + + + + + + 675 + 1594 + 25 + 22 + + + 687.5 + 1605.25 + + + + + + + + Output parameter H + 7772d1f7-9786-4d55-8690-af9d6877a777 + H + H + false + 0 + + + + + + 675 + 1616 + 25 + 23 + + + 687.5 + 1627.75 + + + + + + + + Output parameter A + bc883717-6d8a-41a6-bb0b-3d7c5d7a61ea + A + A + false + 0 + + + + + + 675 + 1639 + 25 + 22 + + + 687.5 + 1650.25 + + + + + + + + Output parameter F + fc8245e0-b2d0-471d-9254-01a0eff7a0b4 + F + F + false + 0 + + + + + + 675 + 1661 + 25 + 23 + + + 687.5 + 1672.75 + + + + + + + + + + + + + + 57da07bd-ecab-415d-9d86-af36d7073abc + Number Slider + + + + + Numeric slider for single values + 7d11a12d-1f6f-4777-9bc7-965dd3035809 + Number Slider + length + false + 0 + + + + + + 163 + 1555 + 382 + 20 + + + 163.9633 + 1555.108 + + + + + + 2 + 1 + 0 + 400 + 0 + 0 + 225 + + + + + + + + + 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Component.Params.IndexOfInputParam(param) + If i > -1 Then + Return Component.Params.Input.ElementAt(i).DataType > 1 ' input parameter DataType of 1 means it's not receiving input (internal or external) + Else + Msg("error", "Input parameter '" & param & "' not found") + Return False + End If + End Function + + Private Sub Msg(ByVal type As String, ByVal msg As String) ' Output an error, warning, or informational message + Select Case type + Case "error" + Component.AddRuntimeMessage(GH_RuntimeMessageLevel.Error, msg) + Print("Error: " & msg) + Case "warning" + Component.AddRuntimeMessage(GH_RuntimeMessageLevel.Warning, msg) + Print("Warning: " & msg) + Case "info" + Component.AddRuntimeMessage(GH_RuntimeMessageLevel.Remark, msg) + Print(msg) + End Select + End Sub + + ' Solve for the m parameter from length and width (reference {1} equation (34), except b = width and K(k) and E(k) should be K(m) and E(m)) + Private Function SolveMFromLenWid(ByVal L As Double, ByVal w As Double) As Double + If w = 0 Then + Return Defined.M_ZERO_W ' for the boundry condition width = 0, bypass the function and return the known m value + End If + + Dim n As Integer = 1 ' Iteration counter (quit if >MAXIT) + Dim lower As Double = 0 ' m must be within this range + Dim upper As Double = 1 + Dim m As Double + Dim cwl As Double + + Do While (upper - lower) > Defined.MAXERR AndAlso (n) < Defined.MAXIT ' Repeat until range narrow enough or MAXIT + m = (upper + lower) / 2 + cwl = 2 * EllipticE(m) / EllipticK(m) - 1 ' calculate w/L with the test value of m + If cwl < w / L Then ' compares the calculated w/L with the actual w/L then narrows the range of possible m + upper = m + Else + lower = m + End If + n += 1 + Loop + Return m + End Function + + ' Solve for the m parameter from length and height (reference {1} equation (33), except K(k) should be K(m) and k = sqrt(m)) + ' Note that it's actually possible to find 2 valid values for m (hence 2 width values) at certain height values + Private Function SolveMFromLenHt(ByVal L As Double, ByVal h As Double) As List(Of Double) + Dim n As Integer = 1 ' Iteration counter (quit if >MAXIT) + Dim lower As Double = 0 ' m must be within this range + Dim upper As Double = 1 + Dim twoWidths As Boolean = h / L >= Defined.DOUBLE_W_HL_RATIO And h / L < Defined.MAX_HL_RATIO ' check to see if h/L is within the range where 2 solutions for the width are possible + Dim m As Double + Dim mult_m As New List(Of Double) + Dim chl As Double + + If twoWidths Then + ' find the first of two possible solutions for m with the following limits: + lower = Defined.M_DOUBLE_W ' see constants at bottom of script + upper = Defined.M_MAXHEIGHT ' see constants at bottom of script + Do While (upper - lower) > Defined.MAXERR AndAlso (n) < Defined.MAXIT ' Repeat until range narrow enough or MAXIT + m = (upper + lower) / 2 + chl = Math.Sqrt(m) / EllipticK(m) ' calculate h/L with the test value of m + If chl > h / L Then ' compares the calculated h/L with the actual h/L then narrows the range of possible m + upper = m + Else + lower = m + End If + n += 1 + Loop + mult_m.Add(m) + + ' then find the second of two possible solutions for m with the following limits: + lower = Defined.M_MAXHEIGHT ' see constants at bottom of script + upper = 1 + Do While (upper - lower) > Defined.MAXERR AndAlso (n) < Defined.MAXIT ' Repeat until range narrow enough or MAXIT + m = (upper + lower) / 2 + chl = Math.Sqrt(m) / EllipticK(m) ' calculate h/L with the test value of m + If chl < h / L Then ' compares the calculated h/L with the actual h/L then narrows the range of possible m + upper = m + Else + lower = m + End If + n += 1 + Loop + + If m <= Defined.M_MAX Then ' return this m parameter only if it falls within the maximum useful value (above which the curve breaks down) + mult_m.Add(m) + End If + + Else + ' find the one possible solution for the m parameter + upper = Defined.M_DOUBLE_W ' limit the upper end of the search to the maximum value of m for which only one solution exists + Do While (upper - lower) > Defined.MAXERR AndAlso (n) < Defined.MAXIT ' Repeat until range narrow enough or MAXIT + m = (upper + lower) / 2 + chl = Math.Sqrt(m) / EllipticK(m) ' calculate h/L with the test value of m + If chl > h / L Then ' compares the calculated h/L with the actual h/L then narrows the range of possible m + upper = m + Else + lower = m + End If + n += 1 + Loop + mult_m.Add(m) + End If + + Return mult_m + End Function + + ' Solve for the m parameter from width and height (derived from reference {1} equations (33) and (34) with same notes as above) + Private Function SolveMFromWidHt(ByVal w As Double, ByVal h As Double) As Double + Dim n As Integer = 1 ' Iteration counter (quit if >MAXIT) + Dim lower As Double = 0 ' m must be within this range + Dim upper As Double = 1 + Dim m As Double + Dim cwh As Double + + Do While (upper - lower) > Defined.MAXERR AndAlso (n) < Defined.MAXIT ' Repeat until range narrow enough or MAXIT + m = (upper + lower) / 2 + cwh = (2 * EllipticE(m) - EllipticK(m)) / Math.Sqrt(m) ' calculate w/h with the test value of m + If cwh < w / h Then ' compares the calculated w/h with the actual w/h then narrows the range of possible m + upper = m + Else + lower = m + End If + n += 1 + Loop + + Return m + End Function + + ' Calculate length based on height and an m parameter, derived from reference {1} equation (33), except K(k) should be K(m) and k = sqrt(m) + Private Function Cal_L(ByVal h As Double, ByVal m As Double) As Double + Return h * EllipticK(m) / Math.Sqrt(m) + End Function + + ' Calculate width based on length and an m parameter, derived from reference {1} equation (34), except b = width and K(k) and E(k) should be K(m) and E(m) + Private Function Cal_W(ByVal L As Double, ByVal m As Double) As Double + Return L * (2 * EllipticE(m) / EllipticK(m) - 1) + End Function + + ' Calculate height based on length and an m parameter, from reference {1} equation (33), except K(k) should be K(m) and k = sqrt(m) + Private Function Cal_H(ByVal L As Double, ByVal m As Double) As Double + Return L * Math.Sqrt(m) / EllipticK(m) + End Function + + ' Calculate the unique m parameter based on a start tangent angle, from reference {2}, just above equation (9a), that states k = Sin(angle / 2 + Pi / 4), + ' but as m = k^2 and due to this script's need for an angle rotated 90Β° versus the one in reference {1}, the following formula is the result + ' New note: verified by reference {4}, pg. 78 at the bottom + Private Function Cal_M(ByVal a As Double) As Double + Return (1 - Math.Cos(a)) / 2 ' equal to Sin^2(a/2) too + End Function + + ' Calculate start tangent angle based on an m parameter, derived from above formula + Private Function Cal_A(ByVal m As Double) As Double + Return Math.Acos(1 - 2 * m) + End Function + + ' This is the heart of this script, taking the found (or specified) length, width, and angle values along with the found m parameter to create + ' a list of points that approximate the shape or form of the elastica. It works by finding the x and y coordinates (which are reversed versus + ' the original equations (12a) and (12b) from reference {2} due to the 90Β° difference in orientation) based on the tangent angle along the curve. + ' See reference {2} for more details on how they derived it. Note that to simplify things, the algorithm only calculates the points for half of the + ' curve, then mirrors those points along the y-axis. + Private Function FindBendForm(ByVal L As Double, ByVal w As Double, ByVal m As Double, ByVal ang As Double, ByVal refPln As Plane) As List(Of Point3d) + L = L / 2 ' because the below algorithm is based on the formulas in reference {2} for only half of the curve + w = w / 2 ' same + + If ang = 0 Then ' if angle (and height) = 0, then simply return the start and end points of the straight line + Dim out As New List(Of Point3d) + out.Add(refPln.PointAt(w, 0, 0)) + out.Add(refPln.PointAt(-w, 0, 0)) + Return out + End If + + Dim x As Double + Dim y As Double + Dim halfCurvePts As New List(Of Point3d) + Dim fullCurvePts As New List(Of Point3d) + Dim translatedPts As New List(Of Point3d) + + ang -= Math.PI / 2 ' a hack to allow this algorithm to work, since the original curve in paper {2} was rotated 90Β° + Dim angB As Double = ang + (-Math.PI / 2 - ang) / Defined.CURVEDIVS ' angB is the 'lowercase theta' which should be in formula {2}(12b) as the interval + ' start [a typo...see equation(3)]. It's necessary to start angB at ang + [interval] instead of just ang due to integration failing at angB = ang + halfCurvePts.Add(New Point3d(w, 0, 0)) ' start with this known initial point, as integration will fail when angB = ang + + ' each point {x, y} is calculated from the tangent angle, angB, that occurs at each point (which is why this iterates from ~ang to -pi/2, the known end condition) + Do While Math.Round(angB, Defined.ROUNDTO) >= Math.Round(-Math.PI / 2, Defined.ROUNDTO) + y = (Math.Sqrt(2) * Math.Sqrt(Math.Sin(ang) - Math.Sin(angB)) * (w + L)) / (2 * EllipticE(m)) ' note that x and y are swapped vs. (12a) and (12b) + x = (L / (Math.Sqrt(2) * EllipticK(m))) * Simpson(angB, -Math.PI / 2, 500, ang) ' calculate the Simpson approximation of the integral (function f below) + ' over the interval angB ('lowercase theta') to -pi/2. side note: is 500 too few iterations for the Simson algorithm? + + If Math.Round(x, Defined.ROUNDTO) = 0 Then x = 0 + halfCurvePts.Add(New Point3d(x, y, 0)) + + angB += (-Math.PI / 2 - ang) / Defined.CURVEDIVS ' onto the next tangent angle + Loop + + ' After finding the x and y values for half of the curve, add the {-x, y} values for the rest of the curve + For Each point As Point3d In halfCurvePts + If Math.Round(point.X, Defined.ROUNDTO) = 0 Then + If Math.Round(point.Y, Defined.ROUNDTO) = 0 Then + fullCurvePts.Add(New Point3d(0, 0, 0)) ' special case when width = 0: when x = 0, only duplicate the point when y = 0 too + End If + Else + fullCurvePts.Add(New Point3d(-point.X, point.Y, 0)) + End If + Next + halfCurvePts.Reverse + fullCurvePts.AddRange(halfCurvePts) + + For Each p As Point3d In fullCurvePts + translatedPts.Add(refPln.PointAt(p.X, p.Y, p.Z)) ' translate the points from the reference plane to the world plane + Next + + Return translatedPts + End Function + + ' Interpolates the points from FindBendForm to create the Elastica curve. Uses start & end tangents for greater accuracy. + Private Function MakeCurve(ByVal pts As List(Of Point3d), ByVal ang As Double, ByVal refPln As Plane) As Curve + If ang <> 0 Then + Dim ts, te As New Vector3d(refPln.XAxis) + ts.Rotate(ang, refPln.ZAxis) + te.Rotate(-ang, refPln.ZAxis) + Return Curve.CreateInterpolatedCurve(pts, 3, CurveKnotStyle.Chord, ts, te) ' 3rd degree curve with 'Chord' Knot Style + Else + Return Curve.CreateInterpolatedCurve(pts, 3) ' if angle (and height) = 0, then simply interpolate the straight line (no start/end tangents) + End If + End Function + + ' Implements the Simpson approximation for an integral of function f below + Public Function Simpson(a As Double, b As Double, n As Integer, theta As Double) As Double 'n should be an even number + Dim j As Integer, s1 As Double, s2 As Double, h As Double + h = (b - a) / n + s1 = 0 + s2 = 0 + For j = 1 To n - 1 Step 2 + s1 = s1 + fn(a + j * h, theta) + Next j + For j = 2 To n - 2 Step 2 + s2 = s2 + fn(a + j * h, theta) + Next j + Simpson = h / 3 * (fn(a, theta) + 4 * s1 + 2 * s2 + fn(b, theta)) + End Function + + ' Specific calculation for the above integration + Public Function fn(x As Double, theta As Double) As Double + fn = Math.Sin(x) / (Math.Sqrt(Math.Sin(theta) - Math.Sin(x))) ' from reference {2} formula (12b) + End Function + + + ' Return the Complete Elliptic integral of the 1st kind + ' Abramowitz and Stegun p.591, formula 17.3.11 + ' Code from http://www.codeproject.com/Articles/566614/Elliptic-integrals + Public Function EllipticK(ByVal m As Double) As Double + Dim sum, term, above, below As Double + sum = 1 + term = 1 + above = 1 + below = 2 + + For i As Integer = 1 To 100 + term *= above / below + sum += Math.Pow(m, i) * Math.Pow(term, 2) + above += 2 + below += 2 + Next + sum *= 0.5 * Math.PI + Return sum + End Function + + + ' Return the Complete Elliptic integral of the 2nd kind + ' Abramowitz and Stegun p.591, formula 17.3.12 + ' Code from http://www.codeproject.com/Articles/566614/Elliptic-integrals + Public Function EllipticE(ByVal m As Double) As Double + Dim sum, term, above, below As Double + sum = 1 + term = 1 + above = 1 + below = 2 + + For i As Integer = 1 To 100 + term *= above / below + sum -= Math.Pow(m, i) * Math.Pow(term, 2) / above + above += 2 + below += 2 + Next + sum *= 0.5 * Math.PI + Return sum + End Function + + Friend Partial NotInheritable Class Defined + Private Sub New() + End Sub + + ' Note: most of these values for m and h/L ratio were found with Wolfram Alpha and either specific intercepts (x=0) or local minima/maxima. They should be constant. + Public Const M_SKETCHY As Double = 0.95 ' value of the m parameter where the curvature near the ends of the curve gets wonky + Public Const M_MAX As Double = 0.993 ' maximum useful value of the m parameter, above which this algorithm for the form of the curve breaks down + Public Const M_ZERO_W As Double = 0.826114765984970336 ' value of the m parameter when width = 0 + Public Const M_MAXHEIGHT As Double = 0.701327460663101223 ' value of the m parameter at maximum possible height of the bent rod/wire + Public Const M_DOUBLE_W As Double = 0.180254422335013983 ' minimum value of the m parameter when two width values are possible for a given height and length + Public Const DOUBLE_W_HL_RATIO As Double = 0.257342117984635757 ' value of the height/length ratio above which there are two possible width values + Public Const MAX_HL_RATIO As Double = 0.403140189705650243 ' maximum possible value of the height/length ratio + + Public Const MAXERR As Double = 0.0000000001 ' error tolerance + Public Const MAXIT As Integer = 100 ' maximum number of iterations + Public Const ROUNDTO As Integer = 10 ' number of decimal places to round off to + Public Const CURVEDIVS As Integer = 50 ' number of sample points for building the curve (or half-curve as it were) + End Class + A VB.NET scriptable component + + 98 + 86 + + true + 15a333e8-a6e9-40f9-ae49-542ab7d2e084 + VB Script + VB + true + 0 + ' ----------------------------------------------------------------- + ' Elastic Bending Script by Will McElwain + ' Created February 2014 + ' + ' DESCRIPTION: + ' This beast creates the so-called 'elastica curve', the shape a long, thin rod or wire makes when it is bent elastically (i.e. not permanently). In this case, force + ' is assumed to only be applied horizontally (which would be in line with the rod at rest) and both ends are assumed to be pinned or hinged meaning they are free + ' to rotate (as opposed to clamped, when the end tangent angle is fixed, usually horizontally). An interesting finding is that it doesn't matter what the material or + ' cross-sectional area is, as long as they're uniform along the entire length. Everything makes the same shape when bent as long as it doesn't cross the threshold + ' from elastic to plastic (permanent) deformation (I don't bother to find that limit here, but can be found if the yield stress for a material is known). + ' + ' Key to the formulas used in this script are elliptic integrals, specifically K(m), the complete elliptic integral of the first kind, and E(m), the complete elliptic + ' integral of the second kind. There was a lot of confusion over the 'm' and 'k' parameters for these functions, as some people use them interchangeably, but they are + ' not the same. m = k^2 (thus k = Sqrt(m)). I try to use the 'm' parameter exclusively to avoid this confusion. Note that there is a unique 'm' parameter for every + ' configuration/shape of the elastica curve. + ' + ' This script tries to find that unique 'm' parameter based on the inputs. The algorithm starts with a test version of m, evaluates an expression, say 2*E(m)/K(m)-1, + ' then compares the result to what it should be (in this case, a known width/length ratio). Iterate until the correct m is found. Once we have m, we can then calculate + ' all of the other unknowns, then find points that lie on that curve, then interpolate those points for the actual curve. You can also use Wolfram|Alpha as I did to + ' find the m parameter based on the equations in this script (example here: http://tiny.cc/t4tpbx for when say width=45.2 and length=67.1). + ' + ' Other notes: + ' * This script works with negative values for width, which will creat a self-intersecting curve (as it should). The curvature of the elastica starts to break down around + ' m=0.95 (~154Β°), but this script will continue to work until M_MAX, m=0.993 (~169Β°). If you wish to ignore self-intersecting curves, set ignoreSelfIntersecting to True + ' * When the only known values are length and height, it is actually possible for certain ratios of height to length to have two valid m values (thus 2 possible widths + ' and angles). This script will return them both. + ' * Only the first two valid parameters (of the required ones) will be used, meaning if all four are connected (length, width or a PtB, height, and angle), this script will + ' only use length and width (or a PtB). + ' * Depending on the magnitude of your inputs (say if they're really small, like if length < 10), you might have to increase the constant ROUNDTO at the bottom + ' + ' REFERENCES: + ' {1} "The elastic rod" by M.E. Pacheco Q. & E. Pina, http://www.scielo.org.mx/pdf/rmfe/v53n2/v53n2a8.pdf + ' {2} "An experiment in nonlinear beam theory" by A. Valiente, http://www.deepdyve.com/lp/doc/I3lwnxdfGz , also here: http://tiny.cc/Valiente_AEiNBT + ' {3} "Snap buckling, writhing and Loop formation In twisted rods" by V.G.A. GOSS, http://myweb.lsbu.ac.uk/~gossga/thesisFinal.pdf + ' {4} "Theory of Elastic Stability" by Stephen Timoshenko, http://www.scribd.com/doc/50402462/Timoshenko-Theory-of-Elastic-Stability (start on p. 76) + ' + ' INPUT: + ' PtA - First anchor point (required) + ' PtB - Second anchor point (optional, though 2 out of the 4--length, width, height, angle--need to be specified) + ' [note that PtB can be the same as PtA (meaning width would be zero)] + ' [also note that if a different width is additionally specified that's not equal to the distance between PtA and PtB, then the end point will not equal PtB anymore] + ' Pln - Plane of the bent rod/wire, which bends up in the +y direction. The line between PtA and PtB (if specified) must be parallel to the x-axis of this plane + ' + ' ** 2 of the following 4 need to be specified ** + ' Len - Length of the rod/wire, which needs to be > 0 + ' Wid - Width between the endpoints of the curve [note: if PtB is specified in addition, and distance between PtA and PtB <> width, the end point will be relocated + ' Ht - Height of the bent rod/wire (when negative, curve will bend downward, relative to the input plane, instead) + ' Ang - Inner departure angle or tangent angle (in radians) at the ends of the bent rod/wire. Set up so as width approaches length (thus height approaches zero), angle approaches zero + ' + ' * Following variables only needed for optional calculating of bending force, not for shape of curve. + ' E - Young's modulus (modulus of elasticity) in GPa (=N/m^2) (material-specific. for example, 7075 aluminum is roughly 71.7 GPa) + ' I - Second moment of area (or area moment of inertia) in m^4 (cross-section-specific. for example, a hollow rod + ' would have I = pi * (outer_diameter^4 - inner_diameter^4) / 32 + ' Note: E*I is also known as flexural rigidity or bending stiffness + ' + ' OUTPUT: + ' out - only for debugging messages + ' Pts - the list of points that approximate the shape of the elastica + ' Crv - the 3rd-degree curve interpolated from those points (with accurate start & end tangents) + ' L - the length of the rod/wire + ' W - the distance (width) between the endpoints of the rod/wire + ' H - the height of the bent rod/wire + ' A - the tangent angle at the (start) end of the rod/wire + ' F - the force needed to hold the rod/wire in a specific shape (based on the material properties & cross-section) **be sure your units for 'I' match your units for the + ' rest of your inputs (length, width, etc.). Also note that the critical buckling load (force) that makes the rod/wire start to bend can be found at height=0 + ' + ' THANKS TO: + ' MΓ₯rten Nettelbladt (thegeometryofbending.blogspot.com) + ' Daniel Piker (Kangaroo plugin) + ' David Rutten (Grasshopper guru) + ' Euler & Bernoulli (the O.G.'s) + ' + ' ----------------------------------------------------------------- + + Dim ignoreSelfIntersecting As Boolean = False ' set to True if you don't want to output curves where width < 0, which creates a self-intersecting curve + + Dim inCt As Integer = 0 ' count the number of required parameters that are receiving data + Dim length As Double + Dim width As System.Object = Nothing ' need to set as Nothing so we can check if it has been assigned a value later + Dim height As Double + Dim angle As Double + Dim m As Double + Dim multiple_m As New List(Of Double) + Dim AtoB As Line + Dim flip_H As Boolean = False ' if height is negative, this flag will be set + Dim flip_A As Boolean = False ' if angle is negative, this flag will be set + + If Not IsSet("Pln") Then + Msg("error", "Base plane is not set") + Return + End If + + If Not IsSet("PtA") Then + Msg("error", "Point A is not set") + Return + End If + + If Math.Round(Pln.DistanceTo(PtA), Defined.ROUNDTO) <> 0 Then + Msg("error", "Point A is not on the base plane") + Return + End If + + Dim refPlane As Plane = Pln ' create a reference plane = input plane and set the origin of it to PtA in case PtA isn't the origin already + refPlane.Origin = PtA + + If IsSet("PtB") Then + If Math.Round(Pln.DistanceTo(PtB), Defined.ROUNDTO) <> 0 Then + Msg("error", "Point B is not on the base plane") + Return + End If + + AtoB = New Line(PtA, PtB) + If AtoB.Length <> 0 And Not AtoB.Direction.IsPerpendicularTo(Pln.YAxis) Then + Msg("error", "The line between PtA and PtB is not perpendicular to the Y-axis of the specified plane") + Return + End If + + inCt += 1 + If IsSet("Wid") Then Msg("info", "Wid will override the distance between PtA and PtB. If you do not want this to happen, disconnect PtB or Wid.") + + width = PtA.DistanceTo(PtB) ' get the width (distance) between PtA and PtB + + Dim refPtB As Point3d + refPlane.RemapToPlaneSpace(PtB, refPtB) + If refPtB.X < 0 Then width = -width ' check if PtB is to the left of PtA...if so, width is negative + End If + + If IsSet("Len") Then inCt += 1 + If IsSet("Wid") Then inCt += 1 + If IsSet("Ht") Then inCt += 1 + If IsSet("Ang") Then inCt += 1 + If inCt > 2 Then Msg("info", "More parameters set than are required (out of length, width, height, angle). Only using the first two valid ones.") + + ' check for connected/specified inputs. note: only the first two that it comes across will be used + If IsSet("Len") Then ' if length is specified then... + If Len <= 0 Then + Msg("error", "Length cannot be negative or zero") + Return + End If + If IsSet("Wid") Then ' find height & angle based on length and specified width + If Wid > Len Then + Msg("error", "Width is greater than length") + Return + End If + If Wid = Len Then ' skip the solver and set the known values + height = 0 + m = 0 + angle = 0 + width = Wid + Else + m = SolveMFromLenWid(Len, Wid) + height = Cal_H(Len, m) ' L * Sqrt(m) / K(m) + angle = Cal_A(m) ' Acos(1 - 2 * m) + width = Wid + End If + + Else If width IsNot Nothing Then ' find height & angle based on length and calculated width (distance between PtA and PtB) + If width > Len Then + Msg("error", "Width is greater than length") + Return + End If + If width = Len Then ' skip the solver and set the known values + height = 0 + m = 0 + angle = 0 + Else + m = SolveMFromLenWid(Len, width) + height = Cal_H(Len, m) ' L * Sqrt(m) / K(m) + angle = Cal_A(m) ' Acos(1 - 2 * m) + End If + + Else If IsSet("Ht") Then ' find width & angle based on length and height ** possible to return 2 results ** + If Math.Abs(Ht / Len) > Defined.MAX_HL_RATIO Then + Msg("error", "Height not possible with given length") + Return + End If + If Ht < 0 Then + Ht = -Ht ' if height is negative, set it to positive (for the calculations) but flip the reference plane about its x-axis + refPlane.Transform(Transform.Mirror(New Plane(refPlane.Origin, refPlane.XAxis, refPlane.ZAxis))) + flip_A = True + flip_H = True + End If + If Ht = 0 Then ' skip the solver and set the known values + width = Len + angle = 0 + Else + multiple_m = SolveMFromLenHt(Len, Ht) ' note that it's possible for two values of m to be found if height is close to max height + If multiple_m.Count = 1 Then ' if there's only one m value returned, calculate the width & angle here. we'll deal with multiple m values later + m = multiple_m.Item(0) + width = Cal_W(Len, m) ' L * (2 * E(m) / K(m) - 1) + angle = Cal_A(m) ' Acos(1 - 2 * m) + End If + End If + height = Ht + + Else If IsSet("Ang") Then ' find width & height based on length and angle + If Ang < 0 Then + Ang = -Ang ' if angle is negative, set it to positive (for the calculations) but flip the reference plane about its x-axis + refPlane.Transform(Transform.Mirror(New Plane(refPlane.Origin, refPlane.XAxis, refPlane.ZAxis))) + flip_A = True + flip_H = True + End If + m = Cal_M(Ang) ' (1 - Cos(a)) / 2 + If Ang = 0 Then ' skip the solver and set the known values + width = Len + height = 0 + Else + width = Cal_W(Len, m) ' L * (2 * E(m) / K(m) - 1) + height = Cal_H(Len, m) ' L * Sqrt(m) / K(m) + End If + angle = Ang + + Else + Msg("error", "Need to specify one more parameter in addition to length") + Return + End If + length = Len + + Else If IsSet("Wid") Then ' if width is specified then... + If IsSet("Ht") Then ' find length & angle based on specified width and height + If Ht < 0 Then + Ht = -Ht ' if height is negative, set it to positive (for the calculations) but flip the reference plane about its x-axis + refPlane.Transform(Transform.Mirror(New Plane(refPlane.Origin, refPlane.XAxis, refPlane.ZAxis))) + flip_A = True + flip_H = True + End If + If Ht = 0 Then ' skip the solver and set the known values + length = Wid + angle = 0 + Else + m = SolveMFromWidHt(Wid, Ht) + length = Cal_L(Ht, m) ' h * K(m) / Sqrt(m) + angle = Cal_A(m) ' Acos(1 - 2 * m) + End If + height = Ht + + Else If IsSet("Ang") Then ' find length & height based on specified width and angle + If Wid = 0 Then + Msg("error", "Curve not possible with width = 0 and an angle as inputs") + Return + End If + If Ang < 0 Then + Ang = -Ang ' if angle is negative, set it to positive (for the calculations) but flip the reference plane about its x-axis + refPlane.Transform(Transform.Mirror(New Plane(refPlane.Origin, refPlane.XAxis, refPlane.ZAxis))) + flip_A = True + flip_H = True + End If + m = Cal_M(Ang) ' (1 - Cos(a)) / 2 + If Ang = 0 Then ' skip the solver and set the known values + length = Wid + height = 0 + Else + length = Wid / (2 * EllipticE(m) / EllipticK(m) - 1) + If length < 0 Then + Msg("error", "Curve not possible at specified width and angle (calculated length is negative)") + Return + End If + height = Cal_H(length, m) ' L * Sqrt(m) / K(m) + End If + angle = Ang + + Else + Msg("error", "Need to specify one more parameter in addition to width (Wid)") + Return + End If + width = Wid + + Else If width IsNot Nothing Then ' if width is determined by PtA and PtB then... + If IsSet("Ht") Then ' find length & angle based on calculated width and height + If Ht < 0 Then + Ht = -Ht ' if height is negative, set it to positive (for the calculations) but flip the reference plane about its x-axis + refPlane.Transform(Transform.Mirror(New Plane(refPlane.Origin, refPlane.XAxis, refPlane.ZAxis))) + flip_A = True + flip_H = True + End If + If Ht = 0 Then ' skip the solver and set the known values + length = width + angle = 0 + Else + m = SolveMFromWidHt(width, Ht) + length = Cal_L(Ht, m) ' h * K(m) / Sqrt(m) + angle = Cal_A(m) ' Acos(1 - 2 * m) + End If + height = Ht + + Else If IsSet("Ang") Then ' find length & height based on calculated width and angle + If width = 0 Then + Msg("error", "Curve not possible with width = 0 and an angle as inputs") + Return + End If + If Ang < 0 Then + Ang = -Ang ' if angle is negative, set it to positive (for the calculations) but flip the reference plane about its x-axis + refPlane.Transform(Transform.Mirror(New Plane(refPlane.Origin, refPlane.XAxis, refPlane.ZAxis))) + flip_A = True + flip_H = True + End If + m = Cal_M(Ang) ' (1 - Cos(a)) / 2 + If Ang = 0 Then ' skip the solver and set the known values + length = width + height = 0 + Else + length = width / (2 * EllipticE(m) / EllipticK(m) - 1) + If length < 0 Then + Msg("error", "Curve not possible at specified width and angle (calculated length is negative)") + Return + End If + height = Cal_H(length, m) ' L * Sqrt(m) / K(m) + End If + angle = Ang + + Else + Msg("error", "Need to specify one more parameter in addition to PtA and PtB") + Return + End If + + Else If IsSet("Ht") Then ' if height is specified then... + If IsSet("Ang") Then ' find length & width based on height and angle + If Ht < 0 Then + Ht = -Ht ' if height is negative, set it to positive (for the calculations) but flip the reference plane about its x-axis + refPlane.Transform(Transform.Mirror(New Plane(refPlane.Origin, refPlane.XAxis, refPlane.ZAxis))) + flip_H = True + flip_A = True + End If + If Ht = 0 Then + Msg("error", "Height can't = 0 if only height and angle are specified") + Return + Else + If Ang < 0 Then + Ang = -Ang ' if angle is negative, set it to positive (for the calculations) but flip the reference plane about its x-axis + refPlane.Transform(Transform.Mirror(New Plane(refPlane.Origin, refPlane.XAxis, refPlane.ZAxis))) + flip_A = Not flip_A + flip_H = Not flip_H + End If + m = Cal_M(Ang) ' (1 - Cos(a)) / 2 + If Ang = 0 Then + Msg("error", "Angle can't = 0 if only height and angle are specified") + Return + Else + length = Cal_L(Ht, m) ' h * K(m) / Sqrt(m) + width = Cal_W(length, m) ' L * (2 * E(m) / K(m) - 1) + End If + angle = Ang + End If + height = Ht + + Else + Msg("error", "Need to specify one more parameter in addition to height") + Return + End If + + Else If IsSet("Ang") Then + Msg("error", "Need to specify one more parameter in addition to angle") + Return + Else + Msg("error", "Need to specify two of the four parameters: length, width (or PtB), height, and angle") + Return + End If + + If m > Defined.M_MAX Then + Msg("error", "Form of curve not solvable with current algorithm and given inputs") + Return + End If + + refPlane.Origin = refPlane.PointAt(width / 2, 0, 0) ' adjust the origin of the reference plane so that the curve is centered about the y-axis (start of the curve is at x = -width/2) + + If multiple_m.Count > 1 Then ' if there is more than one m value returned, calculate the width, angle, and curve for each + Dim multi_pts As New DataTree(Of Point3d) + Dim multi_crv As New List(Of Curve) + Dim tmp_pts As New List(Of Point3d) + Dim multi_W, multi_A, multi_F As New List(Of Double) + Dim j As Integer = 0 ' used for creating a new branch (GH_Path) for storing pts which is itself a list of points + + For Each m_val As Double In multiple_m + width = Cal_W(length, m_val) 'length * (2 * EllipticE(m_val) / EllipticK(m_val) - 1) + + If width < 0 And ignoreSelfIntersecting Then + Msg("warning", "One curve is self-intersecting. To enable these, set ignoreSelfIntersecting to False") + Continue For + End If + + If m_val >= Defined.M_SKETCHY Then Msg("info", "Accuracy of the curve whose width = " & Math.Round(width, 4) & " is not guaranteed") + + angle = Cal_A(m_val) 'Math.Asin(2 * m_val - 1) + refPlane.Origin = refPlane.PointAt(width / 2, 0, 0) ' adjust the origin of the reference plane so that the curve is centered about the y-axis (start of the curve is at x = -width/2) + + tmp_pts = FindBendForm(length, width, m_val, angle, refPlane) + multi_pts.AddRange(tmp_pts, New GH_Path(j)) + multi_crv.Add(MakeCurve(tmp_pts, angle, refPlane)) + + multi_W.Add(width) + If flip_A Then angle = -angle + multi_A.Add(angle) + + E = E * 10 ^ 9 ' Young's modulus input E is in GPa, so we convert to Pa here (= N/m^2) + multi_F.Add(EllipticK(m_val) ^ 2 * E * I / length ^ 2) ' from reference {4} pg. 79 + + j += 1 + refPlane.Origin = PtA ' reset the reference plane origin to PtA for the next m_val + 'Print("length=" & length & ", width=" & width & ", height=" & height & ", angle=" & angle & ", m=" & m_val & ", k=" & Math.Sqrt(m_val) & ", w/L=" & width / length & ", h/L=" & height / length & ", w/h=" & width / height) + Next + + ' assign the outputs + Pts = multi_pts + Crv = multi_crv + L = length + W = multi_W + If flip_H Then height = -height + H = height + A = multi_A + F = multi_F + + Else ' only deal with the single m value + If m >= Defined.M_SKETCHY Then Msg("info", "Accuracy of the curve at these parameters is not guaranteed") + + If width < 0 And ignoreSelfIntersecting Then + Msg("error", "Curve is self-intersecting. To enable these, set ignoreSelfIntersecting to False") + Return + End If + + Pts = FindBendForm(length, width, m, angle, refPlane) + Crv = MakeCurve(pts, angle, refPlane) + L = length + W = width + If flip_H Then height = -height + H = height + If flip_A Then angle = -angle + A = angle + + E = E * 10 ^ 9 ' Young's modulus input E is in GPa, so we convert to Pa here (= N/m^2) + F = EllipticK(m) ^ 2 * E * I / length ^ 2 ' from reference {4} pg. 79. Note: the critical buckling (that makes the rod/wire start to bend) can be found at height=0 (width=length) + + 'height = Math.Sqrt(((2 * Len / 5) ^ 2 - ((Wid - Len / 5) / 2) ^ 2) ' quick approximation discovered by MΓ₯rten of 'Geometry of Bending' fame ( http://tiny.cc/it2pbx ) + 'width = (Len +/- 2 * Math.Sqrt(4 * Len ^ 2 - 25 * Ht ^ 2)) / 5 ' derived from above + 'length = (2 * Math.Sqrt(15 * Ht ^ 2 + 4 * Wid ^ 2) - Wid) / 3 ' derived from above + + 'Print("length=" & length & ", width=" & width & ", height=" & height & ", angle=" & angle & ", m=" & m & ", k=" & Math.Sqrt(m) & ", w/L=" & width / length & ", h/L=" & height / length & ", w/h=" & width / height) + End If + + + + + + + 1797 + 201 + 84 + 184 + + + 1839 + 293 + + + + + + 9 + 84fa917c-1ed8-4db3-8be1-7bdc4a6495a2 + 84fa917c-1ed8-4db3-8be1-7bdc4a6495a2 + 84fa917c-1ed8-4db3-8be1-7bdc4a6495a2 + 84fa917c-1ed8-4db3-8be1-7bdc4a6495a2 + 84fa917c-1ed8-4db3-8be1-7bdc4a6495a2 + 84fa917c-1ed8-4db3-8be1-7bdc4a6495a2 + 84fa917c-1ed8-4db3-8be1-7bdc4a6495a2 + 84fa917c-1ed8-4db3-8be1-7bdc4a6495a2 + 84fa917c-1ed8-4db3-8be1-7bdc4a6495a2 + 8 + 3ede854e-c753-40eb-84cb-b48008f14fd4 + 8ec86459-bf01-4409-baee-174d0d2b13d0 + 8ec86459-bf01-4409-baee-174d0d2b13d0 + 8ec86459-bf01-4409-baee-174d0d2b13d0 + 8ec86459-bf01-4409-baee-174d0d2b13d0 + 8ec86459-bf01-4409-baee-174d0d2b13d0 + 8ec86459-bf01-4409-baee-174d0d2b13d0 + 8ec86459-bf01-4409-baee-174d0d2b13d0 + + + + + true + Script Variable PtA + 512bccc3-6c0e-4ef4-ba24-8685c3ee8d8c + PtA + PtA + true + 0 + true + 1813e6b2-8594-4cdf-882c-e312c60bd7f7 + 1 + e1937b56-b1da-4c12-8bd8-e34ee81746ef + + + + + + 1799 + 203 + 25 + 20 + + + 1813 + 213 + + + + + + + + true + Script Variable PtB + d879c694-aa7a-49cc-885b-4d3c9e0e85df + PtB + PtB + true + 0 + true + 0 + e1937b56-b1da-4c12-8bd8-e34ee81746ef + + + + + + 1799 + 223 + 25 + 20 + + + 1813 + 233 + + + + + + + + true + Script Variable Pln + 0b814522-7d00-47c6-9c48-1e46bee924f2 + Pln + Pln + true + 0 + true + f9c309f7-e784-42bd-ac1d-c6f978935e00 + 1 + 3897522d-58e9-4d60-b38c-978ddacfedd8 + + + + + + 1799 + 243 + 25 + 20 + + + 1813 + 253 + + + + + + + + true + Script Variable Len + 693f977f-077d-410b-a1cc-bc37f0473ad9 + Len + Len + true + 0 + true + ce3bf1e3-3694-43ca-b804-94bf1ac205b6 + 1 + 8e991e99-5fb8-41e1-928d-1bba8fb9f7d7 + + + + + + 1799 + 263 + 25 + 20 + + + 1813 + 273 + + + + + + + + true + Script Variable Wid + e718cabe-f163-44e2-bf0e-4866946c6c49 + Wid + Wid + true + 0 + true + 2dac057e-8756-4d2f-b7af-61904cb5801a + 1 + 8e991e99-5fb8-41e1-928d-1bba8fb9f7d7 + + + + + + 1799 + 283 + 25 + 20 + + + 1813 + 293 + + + + + + + + true + Script Variable Ht + f8027746-ba7f-4f8e-bb9c-fa5b544e826b + Ht + Ht + true + 0 + true + 0 + 8e991e99-5fb8-41e1-928d-1bba8fb9f7d7 + + + + + + 1799 + 303 + 25 + 20 + + + 1813 + 313 + + + + + + + + true + Script Variable Ang + 50106206-1bb3-43fe-bd4f-366e3b16274a + Ang + Ang + true + 0 + true + 0 + 8e991e99-5fb8-41e1-928d-1bba8fb9f7d7 + + + + + + 1799 + 323 + 25 + 20 + + + 1813 + 333 + + + + + + + + true + Script Variable E + 1d6edbeb-7707-43da-8d81-c55c2a788b19 + E + E + true + 0 + true + 0 + 8e991e99-5fb8-41e1-928d-1bba8fb9f7d7 + + + + + + 1799 + 343 + 25 + 20 + + + 1813 + 353 + + + + + + + + true + Script Variable I + 06ae685a-2839-4dc2-a276-de0409a26bad + I + I + true + 0 + true + 0 + 8e991e99-5fb8-41e1-928d-1bba8fb9f7d7 + + + + + + 1799 + 363 + 25 + 20 + + + 1813 + 373 + + + + + + + + 1 + Print, Reflect and Error streams + 20755b5a-7694-4c4e-82c2-fb013dc3fe1b + out + out + false + 0 + + + + + + 1854 + 203 + 25 + 22 + + + 1866.5 + 214.25 + + + + + + + + Output parameter Pts + b6862774-22a9-4557-8df9-4e105338905c + Pts + Pts + false + 0 + + + + + + 1854 + 225 + 25 + 23 + + + 1866.5 + 236.75 + + + + + + + + Output parameter Crv + 8f545f02-550b-41fb-8dd0-70baaad81a72 + Crv + Crv + false + 0 + + + + + + 1854 + 248 + 25 + 22 + + + 1866.5 + 259.25 + + + + + + + + Output parameter L + d7f4a38f-b681-4226-a169-d484336986a2 + L + L + false + 0 + + + + + + 1854 + 270 + 25 + 23 + + + 1866.5 + 281.75 + + + + + + + + Output parameter W + ce2227d2-88d5-44a1-b925-e842136dca13 + W + W + false + 0 + + + + + + 1854 + 293 + 25 + 22 + + + 1866.5 + 304.25 + + + + + + + + Output parameter H + 1c6a682f-bab0-45c4-b876-7f71802d69ab + H + H + false + 0 + + + + + + 1854 + 315 + 25 + 23 + + + 1866.5 + 326.75 + + + + + + + + Output parameter A + 24ef080a-aae1-4bd9-a2eb-97cd1569a733 + A + A + false + 0 + + + + + + 1854 + 338 + 25 + 22 + + + 1866.5 + 349.25 + + + + + + + + Output parameter F + 057d392c-b422-4b34-a8df-30546d6c59e2 + F + F + false + 0 + + + + + + 1854 + 360 + 25 + 23 + + + 1866.5 + 371.75 + + + + + + + + + + + + + + 57da07bd-ecab-415d-9d86-af36d7073abc + Number Slider + + + + + Numeric slider for single values + 2dac057e-8756-4d2f-b7af-61904cb5801a + Number Slider + width + false + 0 + + + + + + 1346 + 281 + 382 + 20 + + + 1346.563 + 281.9091 + + + + + + 2 + 1 + 0 + 400 + -130 + 0 + -43.19 + + + + + + + + + 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+ 0 + + + + + + + + + + + + X-Axis direction of plane + c61582e5-1354-459b-babb-382f6952792a + X-Axis + X + false + a09f5810-c5e6-41cf-a732-76ade1d918e1 + 1 + + + + + + 2676 + 1317 + 14 + 20 + + + 2684.5 + 1327 + + + + + + 1 + + + + + 1 + {0} + + + + + + 1 + 0 + 0 + + + + + + + + + + + + Y-Axis direction of plane + ec988c3e-fd2b-4d2b-aa6a-5e3a9c15a503 + Y-Axis + Y + false + fdc6f14b-d2af-4143-9fa5-98d72e1496f4 + 1 + + + + + + 2676 + 1337 + 14 + 20 + + + 2684.5 + 1347 + + + + + + 1 + + + + + 1 + {0} + + + + + + 0 + 1 + 0 + + + + + + + + + + + + Constructed plane + f5ef3118-8c76-4765-bdd9-16087c3ebb2d + Plane + Pl + false + 0 + + + + + + 2720 + 1297 + 18 + 60 + + + 2729 + 1327 + + + + + + + + + + + + 079bd9bd-54a0-41d4-98af-db999015f63d + VB Script + + + + + Private Function IsSet(ByVal param As String) As Boolean ' Check if an input parameter has data + Dim i As Integer = Component.Params.IndexOfInputParam(param) + If i > -1 Then + Return Component.Params.Input.ElementAt(i).DataType > 1 ' input parameter DataType of 1 means it's not receiving input (internal or external) + Else + Msg("error", "Input parameter '" & param & "' not found") + Return False + End If + End Function + + Private Sub Msg(ByVal type As String, ByVal msg As String) ' Output an error, warning, or informational message + Select Case type + Case "error" + Component.AddRuntimeMessage(GH_RuntimeMessageLevel.Error, msg) + Print("Error: " & msg) + Case "warning" + Component.AddRuntimeMessage(GH_RuntimeMessageLevel.Warning, msg) + Print("Warning: " & msg) + Case "info" + Component.AddRuntimeMessage(GH_RuntimeMessageLevel.Remark, msg) + Print(msg) + End Select + End Sub + + ' Solve for the m parameter from length and width (reference {1} equation (34), except b = width and K(k) and E(k) should be K(m) and E(m)) + Private Function SolveMFromLenWid(ByVal L As Double, ByVal w As Double) As Double + If w = 0 Then + Return Defined.M_ZERO_W ' for the boundry condition width = 0, bypass the function and return the known m value + End If + + Dim n As Integer = 1 ' Iteration counter (quit if >MAXIT) + Dim lower As Double = 0 ' m must be within this range + Dim upper As Double = 1 + Dim m As Double + Dim cwl As Double + + Do While (upper - lower) > Defined.MAXERR AndAlso (n) < Defined.MAXIT ' Repeat until range narrow enough or MAXIT + m = (upper + lower) / 2 + cwl = 2 * EllipticE(m) / EllipticK(m) - 1 ' calculate w/L with the test value of m + If cwl < w / L Then ' compares the calculated w/L with the actual w/L then narrows the range of possible m + upper = m + Else + lower = m + End If + n += 1 + Loop + Return m + End Function + + ' Solve for the m parameter from length and height (reference {1} equation (33), except K(k) should be K(m) and k = sqrt(m)) + ' Note that it's actually possible to find 2 valid values for m (hence 2 width values) at certain height values + Private Function SolveMFromLenHt(ByVal L As Double, ByVal h As Double) As List(Of Double) + Dim n As Integer = 1 ' Iteration counter (quit if >MAXIT) + Dim lower As Double = 0 ' m must be within this range + Dim upper As Double = 1 + Dim twoWidths As Boolean = h / L >= Defined.DOUBLE_W_HL_RATIO And h / L < Defined.MAX_HL_RATIO ' check to see if h/L is within the range where 2 solutions for the width are possible + Dim m As Double + Dim mult_m As New List(Of Double) + Dim chl As Double + + If twoWidths Then + ' find the first of two possible solutions for m with the following limits: + lower = Defined.M_DOUBLE_W ' see constants at bottom of script + upper = Defined.M_MAXHEIGHT ' see constants at bottom of script + Do While (upper - lower) > Defined.MAXERR AndAlso (n) < Defined.MAXIT ' Repeat until range narrow enough or MAXIT + m = (upper + lower) / 2 + chl = Math.Sqrt(m) / EllipticK(m) ' calculate h/L with the test value of m + If chl > h / L Then ' compares the calculated h/L with the actual h/L then narrows the range of possible m + upper = m + Else + lower = m + End If + n += 1 + Loop + mult_m.Add(m) + + ' then find the second of two possible solutions for m with the following limits: + lower = Defined.M_MAXHEIGHT ' see constants at bottom of script + upper = 1 + Do While (upper - lower) > Defined.MAXERR AndAlso (n) < Defined.MAXIT ' Repeat until range narrow enough or MAXIT + m = (upper + lower) / 2 + chl = Math.Sqrt(m) / EllipticK(m) ' calculate h/L with the test value of m + If chl < h / L Then ' compares the calculated h/L with the actual h/L then narrows the range of possible m + upper = m + Else + lower = m + End If + n += 1 + Loop + + If m <= Defined.M_MAX Then ' return this m parameter only if it falls within the maximum useful value (above which the curve breaks down) + mult_m.Add(m) + End If + + Else + ' find the one possible solution for the m parameter + upper = Defined.M_DOUBLE_W ' limit the upper end of the search to the maximum value of m for which only one solution exists + Do While (upper - lower) > Defined.MAXERR AndAlso (n) < Defined.MAXIT ' Repeat until range narrow enough or MAXIT + m = (upper + lower) / 2 + chl = Math.Sqrt(m) / EllipticK(m) ' calculate h/L with the test value of m + If chl > h / L Then ' compares the calculated h/L with the actual h/L then narrows the range of possible m + upper = m + Else + lower = m + End If + n += 1 + Loop + mult_m.Add(m) + End If + + Return mult_m + End Function + + ' Solve for the m parameter from width and height (derived from reference {1} equations (33) and (34) with same notes as above) + Private Function SolveMFromWidHt(ByVal w As Double, ByVal h As Double) As Double + Dim n As Integer = 1 ' Iteration counter (quit if >MAXIT) + Dim lower As Double = 0 ' m must be within this range + Dim upper As Double = 1 + Dim m As Double + Dim cwh As Double + + Do While (upper - lower) > Defined.MAXERR AndAlso (n) < Defined.MAXIT ' Repeat until range narrow enough or MAXIT + m = (upper + lower) / 2 + cwh = (2 * EllipticE(m) - EllipticK(m)) / Math.Sqrt(m) ' calculate w/h with the test value of m + If cwh < w / h Then ' compares the calculated w/h with the actual w/h then narrows the range of possible m + upper = m + Else + lower = m + End If + n += 1 + Loop + + Return m + End Function + + ' Calculate length based on height and an m parameter, derived from reference {1} equation (33), except K(k) should be K(m) and k = sqrt(m) + Private Function Cal_L(ByVal h As Double, ByVal m As Double) As Double + Return h * EllipticK(m) / Math.Sqrt(m) + End Function + + ' Calculate width based on length and an m parameter, derived from reference {1} equation (34), except b = width and K(k) and E(k) should be K(m) and E(m) + Private Function Cal_W(ByVal L As Double, ByVal m As Double) As Double + Return L * (2 * EllipticE(m) / EllipticK(m) - 1) + End Function + + ' Calculate height based on length and an m parameter, from reference {1} equation (33), except K(k) should be K(m) and k = sqrt(m) + Private Function Cal_H(ByVal L As Double, ByVal m As Double) As Double + Return L * Math.Sqrt(m) / EllipticK(m) + End Function + + ' Calculate the unique m parameter based on a start tangent angle, from reference {2}, just above equation (9a), that states k = Sin(angle / 2 + Pi / 4), + ' but as m = k^2 and due to this script's need for an angle rotated 90Β° versus the one in reference {1}, the following formula is the result + ' New note: verified by reference {4}, pg. 78 at the bottom + Private Function Cal_M(ByVal a As Double) As Double + Return (1 - Math.Cos(a)) / 2 ' equal to Sin^2(a/2) too + End Function + + ' Calculate start tangent angle based on an m parameter, derived from above formula + Private Function Cal_A(ByVal m As Double) As Double + Return Math.Acos(1 - 2 * m) + End Function + + ' This is the heart of this script, taking the found (or specified) length, width, and angle values along with the found m parameter to create + ' a list of points that approximate the shape or form of the elastica. It works by finding the x and y coordinates (which are reversed versus + ' the original equations (12a) and (12b) from reference {2} due to the 90Β° difference in orientation) based on the tangent angle along the curve. + ' See reference {2} for more details on how they derived it. Note that to simplify things, the algorithm only calculates the points for half of the + ' curve, then mirrors those points along the y-axis. + Private Function FindBendForm(ByVal L As Double, ByVal w As Double, ByVal m As Double, ByVal ang As Double, ByVal refPln As Plane) As List(Of Point3d) + L = L / 2 ' because the below algorithm is based on the formulas in reference {2} for only half of the curve + w = w / 2 ' same + + If ang = 0 Then ' if angle (and height) = 0, then simply return the start and end points of the straight line + Dim out As New List(Of Point3d) + out.Add(refPln.PointAt(w, 0, 0)) + out.Add(refPln.PointAt(-w, 0, 0)) + Return out + End If + + Dim x As Double + Dim y As Double + Dim halfCurvePts As New List(Of Point3d) + Dim fullCurvePts As New List(Of Point3d) + Dim translatedPts As New List(Of Point3d) + + ang -= Math.PI / 2 ' a hack to allow this algorithm to work, since the original curve in paper {2} was rotated 90Β° + Dim angB As Double = ang + (-Math.PI / 2 - ang) / Defined.CURVEDIVS ' angB is the 'lowercase theta' which should be in formula {2}(12b) as the interval + ' start [a typo...see equation(3)]. It's necessary to start angB at ang + [interval] instead of just ang due to integration failing at angB = ang + halfCurvePts.Add(New Point3d(w, 0, 0)) ' start with this known initial point, as integration will fail when angB = ang + + ' each point {x, y} is calculated from the tangent angle, angB, that occurs at each point (which is why this iterates from ~ang to -pi/2, the known end condition) + Do While Math.Round(angB, Defined.ROUNDTO) >= Math.Round(-Math.PI / 2, Defined.ROUNDTO) + y = (Math.Sqrt(2) * Math.Sqrt(Math.Sin(ang) - Math.Sin(angB)) * (w + L)) / (2 * EllipticE(m)) ' note that x and y are swapped vs. (12a) and (12b) + x = (L / (Math.Sqrt(2) * EllipticK(m))) * Simpson(angB, -Math.PI / 2, 500, ang) ' calculate the Simpson approximation of the integral (function f below) + ' over the interval angB ('lowercase theta') to -pi/2. side note: is 500 too few iterations for the Simson algorithm? + + If Math.Round(x, Defined.ROUNDTO) = 0 Then x = 0 + halfCurvePts.Add(New Point3d(x, y, 0)) + + angB += (-Math.PI / 2 - ang) / Defined.CURVEDIVS ' onto the next tangent angle + Loop + + ' After finding the x and y values for half of the curve, add the {-x, y} values for the rest of the curve + For Each point As Point3d In halfCurvePts + If Math.Round(point.X, Defined.ROUNDTO) = 0 Then + If Math.Round(point.Y, Defined.ROUNDTO) = 0 Then + fullCurvePts.Add(New Point3d(0, 0, 0)) ' special case when width = 0: when x = 0, only duplicate the point when y = 0 too + End If + Else + fullCurvePts.Add(New Point3d(-point.X, point.Y, 0)) + End If + Next + halfCurvePts.Reverse + fullCurvePts.AddRange(halfCurvePts) + + For Each p As Point3d In fullCurvePts + translatedPts.Add(refPln.PointAt(p.X, p.Y, p.Z)) ' translate the points from the reference plane to the world plane + Next + + Return translatedPts + End Function + + ' Interpolates the points from FindBendForm to create the Elastica curve. Uses start & end tangents for greater accuracy. + Private Function MakeCurve(ByVal pts As List(Of Point3d), ByVal ang As Double, ByVal refPln As Plane) As Curve + If ang <> 0 Then + Dim ts, te As New Vector3d(refPln.XAxis) + ts.Rotate(ang, refPln.ZAxis) + te.Rotate(-ang, refPln.ZAxis) + Return Curve.CreateInterpolatedCurve(pts, 3, CurveKnotStyle.Chord, ts, te) ' 3rd degree curve with 'Chord' Knot Style + Else + Return Curve.CreateInterpolatedCurve(pts, 3) ' if angle (and height) = 0, then simply interpolate the straight line (no start/end tangents) + End If + End Function + + ' Implements the Simpson approximation for an integral of function f below + Public Function Simpson(a As Double, b As Double, n As Integer, theta As Double) As Double 'n should be an even number + Dim j As Integer, s1 As Double, s2 As Double, h As Double + h = (b - a) / n + s1 = 0 + s2 = 0 + For j = 1 To n - 1 Step 2 + s1 = s1 + fn(a + j * h, theta) + Next j + For j = 2 To n - 2 Step 2 + s2 = s2 + fn(a + j * h, theta) + Next j + Simpson = h / 3 * (fn(a, theta) + 4 * s1 + 2 * s2 + fn(b, theta)) + End Function + + ' Specific calculation for the above integration + Public Function fn(x As Double, theta As Double) As Double + fn = Math.Sin(x) / (Math.Sqrt(Math.Sin(theta) - Math.Sin(x))) ' from reference {2} formula (12b) + End Function + + + ' Return the Complete Elliptic integral of the 1st kind + ' Abramowitz and Stegun p.591, formula 17.3.11 + ' Code from http://www.codeproject.com/Articles/566614/Elliptic-integrals + Public Function EllipticK(ByVal m As Double) As Double + Dim sum, term, above, below As Double + sum = 1 + term = 1 + above = 1 + below = 2 + + For i As Integer = 1 To 100 + term *= above / below + sum += Math.Pow(m, i) * Math.Pow(term, 2) + above += 2 + below += 2 + Next + sum *= 0.5 * Math.PI + Return sum + End Function + + + ' Return the Complete Elliptic integral of the 2nd kind + ' Abramowitz and Stegun p.591, formula 17.3.12 + ' Code from http://www.codeproject.com/Articles/566614/Elliptic-integrals + Public Function EllipticE(ByVal m As Double) As Double + Dim sum, term, above, below As Double + sum = 1 + term = 1 + above = 1 + below = 2 + + For i As Integer = 1 To 100 + term *= above / below + sum -= Math.Pow(m, i) * Math.Pow(term, 2) / above + above += 2 + below += 2 + Next + sum *= 0.5 * Math.PI + Return sum + End Function + + Friend Partial NotInheritable Class Defined + Private Sub New() + End Sub + + ' Note: most of these values for m and h/L ratio were found with Wolfram Alpha and either specific intercepts (x=0) or local minima/maxima. They should be constant. + Public Const M_SKETCHY As Double = 0.95 ' value of the m parameter where the curvature near the ends of the curve gets wonky + Public Const M_MAX As Double = 0.993 ' maximum useful value of the m parameter, above which this algorithm for the form of the curve breaks down + Public Const M_ZERO_W As Double = 0.826114765984970336 ' value of the m parameter when width = 0 + Public Const M_MAXHEIGHT As Double = 0.701327460663101223 ' value of the m parameter at maximum possible height of the bent rod/wire + Public Const M_DOUBLE_W As Double = 0.180254422335013983 ' minimum value of the m parameter when two width values are possible for a given height and length + Public Const DOUBLE_W_HL_RATIO As Double = 0.257342117984635757 ' value of the height/length ratio above which there are two possible width values + Public Const MAX_HL_RATIO As Double = 0.403140189705650243 ' maximum possible value of the height/length ratio + + Public Const MAXERR As Double = 0.0000000001 ' error tolerance + Public Const MAXIT As Integer = 100 ' maximum number of iterations + Public Const ROUNDTO As Integer = 10 ' number of decimal places to round off to + Public Const CURVEDIVS As Integer = 50 ' number of sample points for building the curve (or half-curve as it were) + End Class + A VB.NET scriptable component + + 98 + 86 + + true + efeac80e-aaa9-43ef-acff-f0dc08a37ca1 + VB Script + VB + true + 0 + ' ----------------------------------------------------------------- + ' Elastic Bending Script by Will McElwain + ' Created February 2014 + ' + ' DESCRIPTION: + ' This beast creates the so-called 'elastica curve', the shape a long, thin rod or wire makes when it is bent elastically (i.e. not permanently). In this case, force + ' is assumed to only be applied horizontally (which would be in line with the rod at rest) and both ends are assumed to be pinned or hinged meaning they are free + ' to rotate (as opposed to clamped, when the end tangent angle is fixed, usually horizontally). An interesting finding is that it doesn't matter what the material or + ' cross-sectional area is, as long as they're uniform along the entire length. Everything makes the same shape when bent as long as it doesn't cross the threshold + ' from elastic to plastic (permanent) deformation (I don't bother to find that limit here, but can be found if the yield stress for a material is known). + ' + ' Key to the formulas used in this script are elliptic integrals, specifically K(m), the complete elliptic integral of the first kind, and E(m), the complete elliptic + ' integral of the second kind. There was a lot of confusion over the 'm' and 'k' parameters for these functions, as some people use them interchangeably, but they are + ' not the same. m = k^2 (thus k = Sqrt(m)). I try to use the 'm' parameter exclusively to avoid this confusion. Note that there is a unique 'm' parameter for every + ' configuration/shape of the elastica curve. + ' + ' This script tries to find that unique 'm' parameter based on the inputs. The algorithm starts with a test version of m, evaluates an expression, say 2*E(m)/K(m)-1, + ' then compares the result to what it should be (in this case, a known width/length ratio). Iterate until the correct m is found. Once we have m, we can then calculate + ' all of the other unknowns, then find points that lie on that curve, then interpolate those points for the actual curve. You can also use Wolfram|Alpha as I did to + ' find the m parameter based on the equations in this script (example here: http://tiny.cc/t4tpbx for when say width=45.2 and length=67.1). + ' + ' Other notes: + ' * This script works with negative values for width, which will creat a self-intersecting curve (as it should). The curvature of the elastica starts to break down around + ' m=0.95 (~154Β°), but this script will continue to work until M_MAX, m=0.993 (~169Β°). If you wish to ignore self-intersecting curves, set ignoreSelfIntersecting to True + ' * When the only known values are length and height, it is actually possible for certain ratios of height to length to have two valid m values (thus 2 possible widths + ' and angles). This script will return them both. + ' * Only the first two valid parameters (of the required ones) will be used, meaning if all four are connected (length, width or a PtB, height, and angle), this script will + ' only use length and width (or a PtB). + ' * Depending on the magnitude of your inputs (say if they're really small, like if length < 10), you might have to increase the constant ROUNDTO at the bottom + ' + ' REFERENCES: + ' {1} "The elastic rod" by M.E. Pacheco Q. & E. Pina, http://www.scielo.org.mx/pdf/rmfe/v53n2/v53n2a8.pdf + ' {2} "An experiment in nonlinear beam theory" by A. Valiente, http://www.deepdyve.com/lp/doc/I3lwnxdfGz , also here: http://tiny.cc/Valiente_AEiNBT + ' {3} "Snap buckling, writhing and Loop formation In twisted rods" by V.G.A. GOSS, http://myweb.lsbu.ac.uk/~gossga/thesisFinal.pdf + ' {4} "Theory of Elastic Stability" by Stephen Timoshenko, http://www.scribd.com/doc/50402462/Timoshenko-Theory-of-Elastic-Stability (start on p. 76) + ' + ' INPUT: + ' PtA - First anchor point (required) + ' PtB - Second anchor point (optional, though 2 out of the 4--length, width, height, angle--need to be specified) + ' [note that PtB can be the same as PtA (meaning width would be zero)] + ' [also note that if a different width is additionally specified that's not equal to the distance between PtA and PtB, then the end point will not equal PtB anymore] + ' Pln - Plane of the bent rod/wire, which bends up in the +y direction. The line between PtA and PtB (if specified) must be parallel to the x-axis of this plane + ' + ' ** 2 of the following 4 need to be specified ** + ' Len - Length of the rod/wire, which needs to be > 0 + ' Wid - Width between the endpoints of the curve [note: if PtB is specified in addition, and distance between PtA and PtB <> width, the end point will be relocated + ' Ht - Height of the bent rod/wire (when negative, curve will bend downward, relative to the input plane, instead) + ' Ang - Inner departure angle or tangent angle (in radians) at the ends of the bent rod/wire. Set up so as width approaches length (thus height approaches zero), angle approaches zero + ' + ' * Following variables only needed for optional calculating of bending force, not for shape of curve. + ' E - Young's modulus (modulus of elasticity) in GPa (=N/m^2) (material-specific. for example, 7075 aluminum is roughly 71.7 GPa) + ' I - Second moment of area (or area moment of inertia) in m^4 (cross-section-specific. for example, a hollow rod + ' would have I = pi * (outer_diameter^4 - inner_diameter^4) / 32 + ' Note: E*I is also known as flexural rigidity or bending stiffness + ' + ' OUTPUT: + ' out - only for debugging messages + ' Pts - the list of points that approximate the shape of the elastica + ' Crv - the 3rd-degree curve interpolated from those points (with accurate start & end tangents) + ' L - the length of the rod/wire + ' W - the distance (width) between the endpoints of the rod/wire + ' H - the height of the bent rod/wire + ' A - the tangent angle at the (start) end of the rod/wire + ' F - the force needed to hold the rod/wire in a specific shape (based on the material properties & cross-section) **be sure your units for 'I' match your units for the + ' rest of your inputs (length, width, etc.). Also note that the critical buckling load (force) that makes the rod/wire start to bend can be found at height=0 + ' + ' THANKS TO: + ' MΓ₯rten Nettelbladt (thegeometryofbending.blogspot.com) + ' Daniel Piker (Kangaroo plugin) + ' David Rutten (Grasshopper guru) + ' Euler & Bernoulli (the O.G.'s) + ' + ' ----------------------------------------------------------------- + + Dim ignoreSelfIntersecting As Boolean = False ' set to True if you don't want to output curves where width < 0, which creates a self-intersecting curve + + Dim inCt As Integer = 0 ' count the number of required parameters that are receiving data + Dim length As Double + Dim width As System.Object = Nothing ' need to set as Nothing so we can check if it has been assigned a value later + Dim height As Double + Dim angle As Double + Dim m As Double + Dim multiple_m As New List(Of Double) + Dim AtoB As Line + Dim flip_H As Boolean = False ' if height is negative, this flag will be set + Dim flip_A As Boolean = False ' if angle is negative, this flag will be set + + If Not IsSet("Pln") Then + Msg("error", "Base plane is not set") + Return + End If + + If Not IsSet("PtA") Then + Msg("error", "Point A is not set") + Return + End If + + If Math.Round(Pln.DistanceTo(PtA), Defined.ROUNDTO) <> 0 Then + Msg("error", "Point A is not on the base plane") + Return + End If + + Dim refPlane As Plane = Pln ' create a reference plane = input plane and set the origin of it to PtA in case PtA isn't the origin already + refPlane.Origin = PtA + + If IsSet("PtB") Then + If Math.Round(Pln.DistanceTo(PtB), Defined.ROUNDTO) <> 0 Then + Msg("error", "Point B is not on the base plane") + Return + End If + + AtoB = New Line(PtA, PtB) + If AtoB.Length <> 0 And Not AtoB.Direction.IsPerpendicularTo(Pln.YAxis) Then + Msg("error", "The line between PtA and PtB is not perpendicular to the Y-axis of the specified plane") + Return + End If + + inCt += 1 + If IsSet("Wid") Then Msg("info", "Wid will override the distance between PtA and PtB. If you do not want this to happen, disconnect PtB or Wid.") + + width = PtA.DistanceTo(PtB) ' get the width (distance) between PtA and PtB + + Dim refPtB As Point3d + refPlane.RemapToPlaneSpace(PtB, refPtB) + If refPtB.X < 0 Then width = -width ' check if PtB is to the left of PtA...if so, width is negative + End If + + If IsSet("Len") Then inCt += 1 + If IsSet("Wid") Then inCt += 1 + If IsSet("Ht") Then inCt += 1 + If IsSet("Ang") Then inCt += 1 + If inCt > 2 Then Msg("info", "More parameters set than are required (out of length, width, height, angle). Only using the first two valid ones.") + + ' check for connected/specified inputs. note: only the first two that it comes across will be used + If IsSet("Len") Then ' if length is specified then... + If Len <= 0 Then + Msg("error", "Length cannot be negative or zero") + Return + End If + If IsSet("Wid") Then ' find height & angle based on length and specified width + If Wid > Len Then + Msg("error", "Width is greater than length") + Return + End If + If Wid = Len Then ' skip the solver and set the known values + height = 0 + m = 0 + angle = 0 + width = Wid + Else + m = SolveMFromLenWid(Len, Wid) + height = Cal_H(Len, m) ' L * Sqrt(m) / K(m) + angle = Cal_A(m) ' Acos(1 - 2 * m) + width = Wid + End If + + Else If width IsNot Nothing Then ' find height & angle based on length and calculated width (distance between PtA and PtB) + If width > Len Then + Msg("error", "Width is greater than length") + Return + End If + If width = Len Then ' skip the solver and set the known values + height = 0 + m = 0 + angle = 0 + Else + m = SolveMFromLenWid(Len, width) + height = Cal_H(Len, m) ' L * Sqrt(m) / K(m) + angle = Cal_A(m) ' Acos(1 - 2 * m) + End If + + Else If IsSet("Ht") Then ' find width & angle based on length and height ** possible to return 2 results ** + If Math.Abs(Ht / Len) > Defined.MAX_HL_RATIO Then + Msg("error", "Height not possible with given length") + Return + End If + If Ht < 0 Then + Ht = -Ht ' if height is negative, set it to positive (for the calculations) but flip the reference plane about its x-axis + refPlane.Transform(Transform.Mirror(New Plane(refPlane.Origin, refPlane.XAxis, refPlane.ZAxis))) + flip_A = True + flip_H = True + End If + If Ht = 0 Then ' skip the solver and set the known values + width = Len + angle = 0 + Else + multiple_m = SolveMFromLenHt(Len, Ht) ' note that it's possible for two values of m to be found if height is close to max height + If multiple_m.Count = 1 Then ' if there's only one m value returned, calculate the width & angle here. we'll deal with multiple m values later + m = multiple_m.Item(0) + width = Cal_W(Len, m) ' L * (2 * E(m) / K(m) - 1) + angle = Cal_A(m) ' Acos(1 - 2 * m) + End If + End If + height = Ht + + Else If IsSet("Ang") Then ' find width & height based on length and angle + If Ang < 0 Then + Ang = -Ang ' if angle is negative, set it to positive (for the calculations) but flip the reference plane about its x-axis + refPlane.Transform(Transform.Mirror(New Plane(refPlane.Origin, refPlane.XAxis, refPlane.ZAxis))) + flip_A = True + flip_H = True + End If + m = Cal_M(Ang) ' (1 - Cos(a)) / 2 + If Ang = 0 Then ' skip the solver and set the known values + width = Len + height = 0 + Else + width = Cal_W(Len, m) ' L * (2 * E(m) / K(m) - 1) + height = Cal_H(Len, m) ' L * Sqrt(m) / K(m) + End If + angle = Ang + + Else + Msg("error", "Need to specify one more parameter in addition to length") + Return + End If + length = Len + + Else If IsSet("Wid") Then ' if width is specified then... + If IsSet("Ht") Then ' find length & angle based on specified width and height + If Ht < 0 Then + Ht = -Ht ' if height is negative, set it to positive (for the calculations) but flip the reference plane about its x-axis + refPlane.Transform(Transform.Mirror(New Plane(refPlane.Origin, refPlane.XAxis, refPlane.ZAxis))) + flip_A = True + flip_H = True + End If + If Ht = 0 Then ' skip the solver and set the known values + length = Wid + angle = 0 + Else + m = SolveMFromWidHt(Wid, Ht) + length = Cal_L(Ht, m) ' h * K(m) / Sqrt(m) + angle = Cal_A(m) ' Acos(1 - 2 * m) + End If + height = Ht + + Else If IsSet("Ang") Then ' find length & height based on specified width and angle + If Wid = 0 Then + Msg("error", "Curve not possible with width = 0 and an angle as inputs") + Return + End If + If Ang < 0 Then + Ang = -Ang ' if angle is negative, set it to positive (for the calculations) but flip the reference plane about its x-axis + refPlane.Transform(Transform.Mirror(New Plane(refPlane.Origin, refPlane.XAxis, refPlane.ZAxis))) + flip_A = True + flip_H = True + End If + m = Cal_M(Ang) ' (1 - Cos(a)) / 2 + If Ang = 0 Then ' skip the solver and set the known values + length = Wid + height = 0 + Else + length = Wid / (2 * EllipticE(m) / EllipticK(m) - 1) + If length < 0 Then + Msg("error", "Curve not possible at specified width and angle (calculated length is negative)") + Return + End If + height = Cal_H(length, m) ' L * Sqrt(m) / K(m) + End If + angle = Ang + + Else + Msg("error", "Need to specify one more parameter in addition to width (Wid)") + Return + End If + width = Wid + + Else If width IsNot Nothing Then ' if width is determined by PtA and PtB then... + If IsSet("Ht") Then ' find length & angle based on calculated width and height + If Ht < 0 Then + Ht = -Ht ' if height is negative, set it to positive (for the calculations) but flip the reference plane about its x-axis + refPlane.Transform(Transform.Mirror(New Plane(refPlane.Origin, refPlane.XAxis, refPlane.ZAxis))) + flip_A = True + flip_H = True + End If + If Ht = 0 Then ' skip the solver and set the known values + length = width + angle = 0 + Else + m = SolveMFromWidHt(width, Ht) + length = Cal_L(Ht, m) ' h * K(m) / Sqrt(m) + angle = Cal_A(m) ' Acos(1 - 2 * m) + End If + height = Ht + + Else If IsSet("Ang") Then ' find length & height based on calculated width and angle + If width = 0 Then + Msg("error", "Curve not possible with width = 0 and an angle as inputs") + Return + End If + If Ang < 0 Then + Ang = -Ang ' if angle is negative, set it to positive (for the calculations) but flip the reference plane about its x-axis + refPlane.Transform(Transform.Mirror(New Plane(refPlane.Origin, refPlane.XAxis, refPlane.ZAxis))) + flip_A = True + flip_H = True + End If + m = Cal_M(Ang) ' (1 - Cos(a)) / 2 + If Ang = 0 Then ' skip the solver and set the known values + length = width + height = 0 + Else + length = width / (2 * EllipticE(m) / EllipticK(m) - 1) + If length < 0 Then + Msg("error", "Curve not possible at specified width and angle (calculated length is negative)") + Return + End If + height = Cal_H(length, m) ' L * Sqrt(m) / K(m) + End If + angle = Ang + + Else + Msg("error", "Need to specify one more parameter in addition to PtA and PtB") + Return + End If + + Else If IsSet("Ht") Then ' if height is specified then... + If IsSet("Ang") Then ' find length & width based on height and angle + If Ht < 0 Then + Ht = -Ht ' if height is negative, set it to positive (for the calculations) but flip the reference plane about its x-axis + refPlane.Transform(Transform.Mirror(New Plane(refPlane.Origin, refPlane.XAxis, refPlane.ZAxis))) + flip_H = True + flip_A = True + End If + If Ht = 0 Then + Msg("error", "Height can't = 0 if only height and angle are specified") + Return + Else + If Ang < 0 Then + Ang = -Ang ' if angle is negative, set it to positive (for the calculations) but flip the reference plane about its x-axis + refPlane.Transform(Transform.Mirror(New Plane(refPlane.Origin, refPlane.XAxis, refPlane.ZAxis))) + flip_A = Not flip_A + flip_H = Not flip_H + End If + m = Cal_M(Ang) ' (1 - Cos(a)) / 2 + If Ang = 0 Then + Msg("error", "Angle can't = 0 if only height and angle are specified") + Return + Else + length = Cal_L(Ht, m) ' h * K(m) / Sqrt(m) + width = Cal_W(length, m) ' L * (2 * E(m) / K(m) - 1) + End If + angle = Ang + End If + height = Ht + + Else + Msg("error", "Need to specify one more parameter in addition to height") + Return + End If + + Else If IsSet("Ang") Then + Msg("error", "Need to specify one more parameter in addition to angle") + Return + Else + Msg("error", "Need to specify two of the four parameters: length, width (or PtB), height, and angle") + Return + End If + + If m > Defined.M_MAX Then + Msg("error", "Form of curve not solvable with current algorithm and given inputs") + Return + End If + + refPlane.Origin = refPlane.PointAt(width / 2, 0, 0) ' adjust the origin of the reference plane so that the curve is centered about the y-axis (start of the curve is at x = -width/2) + + If multiple_m.Count > 1 Then ' if there is more than one m value returned, calculate the width, angle, and curve for each + Dim multi_pts As New DataTree(Of Point3d) + Dim multi_crv As New List(Of Curve) + Dim tmp_pts As New List(Of Point3d) + Dim multi_W, multi_A, multi_F As New List(Of Double) + Dim j As Integer = 0 ' used for creating a new branch (GH_Path) for storing pts which is itself a list of points + + For Each m_val As Double In multiple_m + width = Cal_W(length, m_val) 'length * (2 * EllipticE(m_val) / EllipticK(m_val) - 1) + + If width < 0 And ignoreSelfIntersecting Then + Msg("warning", "One curve is self-intersecting. To enable these, set ignoreSelfIntersecting to False") + Continue For + End If + + If m_val >= Defined.M_SKETCHY Then Msg("info", "Accuracy of the curve whose width = " & Math.Round(width, 4) & " is not guaranteed") + + angle = Cal_A(m_val) 'Math.Asin(2 * m_val - 1) + refPlane.Origin = refPlane.PointAt(width / 2, 0, 0) ' adjust the origin of the reference plane so that the curve is centered about the y-axis (start of the curve is at x = -width/2) + + tmp_pts = FindBendForm(length, width, m_val, angle, refPlane) + multi_pts.AddRange(tmp_pts, New GH_Path(j)) + multi_crv.Add(MakeCurve(tmp_pts, angle, refPlane)) + + multi_W.Add(width) + If flip_A Then angle = -angle + multi_A.Add(angle) + + E = E * 10 ^ 9 ' Young's modulus input E is in GPa, so we convert to Pa here (= N/m^2) + multi_F.Add(EllipticK(m_val) ^ 2 * E * I / length ^ 2) ' from reference {4} pg. 79 + + j += 1 + refPlane.Origin = PtA ' reset the reference plane origin to PtA for the next m_val + 'Print("length=" & length & ", width=" & width & ", height=" & height & ", angle=" & angle & ", m=" & m_val & ", k=" & Math.Sqrt(m_val) & ", w/L=" & width / length & ", h/L=" & height / length & ", w/h=" & width / height) + Next + + ' assign the outputs + Pts = multi_pts + Crv = multi_crv + L = length + W = multi_W + If flip_H Then height = -height + H = height + A = multi_A + F = multi_F + + Else ' only deal with the single m value + If m >= Defined.M_SKETCHY Then Msg("info", "Accuracy of the curve at these parameters is not guaranteed") + + If width < 0 And ignoreSelfIntersecting Then + Msg("error", "Curve is self-intersecting. To enable these, set ignoreSelfIntersecting to False") + Return + End If + + Pts = FindBendForm(length, width, m, angle, refPlane) + Crv = MakeCurve(pts, angle, refPlane) + L = length + W = width + If flip_H Then height = -height + H = height + If flip_A Then angle = -angle + A = angle + + E = E * 10 ^ 9 ' Young's modulus input E is in GPa, so we convert to Pa here (= N/m^2) + F = EllipticK(m) ^ 2 * E * I / length ^ 2 ' from reference {4} pg. 79. Note: the critical buckling (that makes the rod/wire start to bend) can be found at height=0 (width=length) + + 'height = Math.Sqrt(((2 * Len / 5) ^ 2 - ((Wid - Len / 5) / 2) ^ 2) ' quick approximation discovered by MΓ₯rten of 'Geometry of Bending' fame ( http://tiny.cc/it2pbx ) + 'width = (Len +/- 2 * Math.Sqrt(4 * Len ^ 2 - 25 * Ht ^ 2)) / 5 ' derived from above + 'length = (2 * Math.Sqrt(15 * Ht ^ 2 + 4 * Wid ^ 2) - Wid) / 3 ' derived from above + + 'Print("length=" & length & ", width=" & width & ", height=" & height & ", angle=" & angle & ", m=" & m & ", k=" & Math.Sqrt(m) & ", w/L=" & width / length & ", h/L=" & height / length & ", w/h=" & width / height) + End If + + + + + + + 2806 + 1380 + 84 + 184 + + + 2848 + 1472 + + + + + + 9 + 84fa917c-1ed8-4db3-8be1-7bdc4a6495a2 + 84fa917c-1ed8-4db3-8be1-7bdc4a6495a2 + 84fa917c-1ed8-4db3-8be1-7bdc4a6495a2 + 84fa917c-1ed8-4db3-8be1-7bdc4a6495a2 + 84fa917c-1ed8-4db3-8be1-7bdc4a6495a2 + 84fa917c-1ed8-4db3-8be1-7bdc4a6495a2 + 84fa917c-1ed8-4db3-8be1-7bdc4a6495a2 + 84fa917c-1ed8-4db3-8be1-7bdc4a6495a2 + 84fa917c-1ed8-4db3-8be1-7bdc4a6495a2 + 8 + 3ede854e-c753-40eb-84cb-b48008f14fd4 + 8ec86459-bf01-4409-baee-174d0d2b13d0 + 8ec86459-bf01-4409-baee-174d0d2b13d0 + 8ec86459-bf01-4409-baee-174d0d2b13d0 + 8ec86459-bf01-4409-baee-174d0d2b13d0 + 8ec86459-bf01-4409-baee-174d0d2b13d0 + 8ec86459-bf01-4409-baee-174d0d2b13d0 + 8ec86459-bf01-4409-baee-174d0d2b13d0 + + + + + true + Script Variable PtA + 89fe7d9c-b998-43b5-8788-30064fd07e47 + PtA + PtA + true + 0 + true + c54e879a-bb6c-47c6-b366-aaa5a16a426a + 83dc0e24-1c12-445e-8647-dffc452baa6a + 2 + e1937b56-b1da-4c12-8bd8-e34ee81746ef + + + + + + 2808 + 1382 + 25 + 20 + + + 2822 + 1392 + + + + + + + + true + Script Variable PtB + d6635d82-91c7-476c-acf5-3f2feae5d91b + PtB + PtB + true + 0 + true + 72b7092f-b51c-4a8b-b457-5cc6f58de91d + 5c0c8191-e5d5-44d8-9b74-ad7b50095c12 + 2 + e1937b56-b1da-4c12-8bd8-e34ee81746ef + + + + + + 2808 + 1402 + 25 + 20 + + + 2822 + 1412 + + + + + + + + true + Script Variable Pln + 8a5ea10d-8baa-47cd-8970-989b44d90853 + Pln + Pln + true + 0 + true + f5ef3118-8c76-4765-bdd9-16087c3ebb2d + 1 + 3897522d-58e9-4d60-b38c-978ddacfedd8 + + + + + + 2808 + 1422 + 25 + 20 + + + 2822 + 1432 + + + + + + + + true + Script Variable Len + 4adfaa55-4523-43d7-9887-fcfc9daa5d32 + Len + Len + true + 0 + true + 0 + 8e991e99-5fb8-41e1-928d-1bba8fb9f7d7 + + + + + + 2808 + 1442 + 25 + 20 + + + 2822 + 1452 + + + + + + + + true + Script Variable Wid + 8d352101-30bc-4590-8ef7-3bebef383668 + Wid + Wid + true + 0 + true + 0 + 8e991e99-5fb8-41e1-928d-1bba8fb9f7d7 + + + + + + 2808 + 1462 + 25 + 20 + + + 2822 + 1472 + + + + + + + + true + Script Variable Ht + ed97f943-b85a-4bc0-84db-64b077475aca + Ht + Ht + true + 0 + true + cd2f5af0-b741-48d5-8bbd-e9695d5278d8 + 1 + 8e991e99-5fb8-41e1-928d-1bba8fb9f7d7 + + + + + + 2808 + 1482 + 25 + 20 + + + 2822 + 1492 + + + + + + + + true + Script Variable Ang + c30d75f7-bb99-4be5-88aa-f992b8386696 + Ang + Ang + true + 0 + true + 0 + 8e991e99-5fb8-41e1-928d-1bba8fb9f7d7 + + + + + + 2808 + 1502 + 25 + 20 + + + 2822 + 1512 + + + + + + + + true + Script Variable E + 33bfc033-c0d3-4ebb-a5b0-659df3a0b51e + E + E + true + 0 + true + 0 + 8e991e99-5fb8-41e1-928d-1bba8fb9f7d7 + + + + + + 2808 + 1522 + 25 + 20 + + + 2822 + 1532 + + + + + + + + true + Script Variable I + 66f1e54b-d2c1-43dd-8716-e6fa7a1ddeca + I + I + true + 0 + true + 0 + 8e991e99-5fb8-41e1-928d-1bba8fb9f7d7 + + + + + + 2808 + 1542 + 25 + 20 + + + 2822 + 1552 + + + + + + + + 1 + Print, Reflect and Error streams + be85f9e8-bde2-44f9-9bb8-80b1010c9d68 + out + out + false + 0 + + + + + + 2863 + 1382 + 25 + 22 + + + 2875.5 + 1393.25 + + + + + + + + Output parameter Pts + bacf233a-c717-41ce-b241-8552418f09f9 + Pts + Pts + false + 0 + + + + + + 2863 + 1404 + 25 + 23 + + + 2875.5 + 1415.75 + + + + + + + + Output parameter Crv + 4fa97325-f527-43c0-a006-78cf944c5e40 + Crv + Crv + false + 0 + + + + + + 2863 + 1427 + 25 + 22 + + + 2875.5 + 1438.25 + + + + + + + + Output parameter L + b9ed95d2-8f4a-49e9-a734-c73e21c200b4 + L + L + false + 0 + + + + + + 2863 + 1449 + 25 + 23 + + + 2875.5 + 1460.75 + + + + + + + + Output parameter W + 264cbae0-f043-490d-9121-0e45630a0b2f + W + W + false + 0 + + + + + + 2863 + 1472 + 25 + 22 + + + 2875.5 + 1483.25 + + + + + + + + Output parameter H + 1578bf3a-8211-4a77-85d0-3cd951789283 + H + H + false + 0 + + + + + + 2863 + 1494 + 25 + 23 + + + 2875.5 + 1505.75 + + + + + + + + Output parameter A + 97c17859-cbbf-4962-a448-d8665c3e706a + A + A + false + 0 + + + + + + 2863 + 1517 + 25 + 22 + + + 2875.5 + 1528.25 + + + + + + + + Output parameter F + 8f3ba33f-2383-4ae9-b911-b9b6e4553a35 + F + F + false + 0 + + + + + + 2863 + 1539 + 25 + 23 + + + 2875.5 + 1550.75 + + + + + + + + + + + + + + c277f778-6fdf-4890-8f78-347efb23c406 + Pipe + + + + + Create a pipe surface around a rail curve. + a18b4542-f317-4532-9634-9ef245df4e2c + Pipe + Pipe + + + + + + 3076 + 1319 + 64 + 64 + + + 3107 + 1351 + + + + + + Base curve + 9be76020-39f2-465f-b508-f09d3f48a425 + Curve + C + false + 4fa97325-f527-43c0-a006-78cf944c5e40 + 1 + + + + + + 3078 + 1321 + 14 + 20 + + + 3086.5 + 1331 + + + + + + + + Pipe radius + 0f317bff-3048-480b-865a-13a2e1cb5106 + Radius + R + false + f2abb0db-802c-4f59-83cb-393711b4a3d9 + 1 + + + + + + 3078 + 1341 + 14 + 20 + + + 3086.5 + 1351 + + + + + + 1 + + + + + 1 + {0} + + + + + 1 + + + + + + + + + + + Specifies the type of caps (0=None, 1=Flat, 2=Round) + 38981e04-97f9-4736-a46b-c55825f5b8fe + Caps + E + false + 0 + + + + + + 3078 + 1361 + 14 + 20 + + + 3086.5 + 1371 + + + + + + 1 + + + + + 1 + {0} + + + + + 0 + + + + + + + + + + + 1 + Resulting Pipe + d118f269-0338-4d1d-b06d-018819a25f1a + Pipe + P + false + 0 + + + + + + 3122 + 1321 + 16 + 60 + + + 3130 + 1351 + + + + + + + + + + + + 59e0b89a-e487-49f8-bab8-b5bab16be14c + Panel + + + + + A panel for custom notes and text values + f2abb0db-802c-4f59-83cb-393711b4a3d9 + Panel + + false + 0 + 0 + .25 + + + + + + 2955 + 1314 + 50 + 20 + + 0 + 0 + 0 + + 2955.289 + 1314.952 + + + + + + + 255;255;250;90 + + true + true + true + false + false + true + + + + + + + + + 59e0b89a-e487-49f8-bab8-b5bab16be14c + Panel + + + + + A panel for custom notes and text values + 26ed0e55-cf0a-436e-9447-805704984e62 + Panel + + false + 0 + 087cd096-ba72-4a81-a462-a9ce3ce43ec4 + 1 + Double click to edit panel content… + + + + + + 3045 + 1553 + 105 + 55 + + 0 + 0 + 0 + + 3045.485 + 1553.607 + + + + + + + 255;255;250;90 + + true + true + true + false + false + true + + + + + + + + + 0d77c51e-584f-44e8-aed2-c2ddf4803888 + Degrees + + + + + Convert an angle specified in radians to degrees + bb620c76-8624-412c-8f17-4aa82e09f1b9 + Degrees + Deg + + + + + + 2955 + 1562 + 64 + 28 + + + 2985 + 1576 + + + + + + Angle in radians + e10ae9f2-834e-4fd3-afbb-6416a8f9ca28 + Radians + R + false + 97c17859-cbbf-4962-a448-d8665c3e706a + 1 + + + + + + 2957 + 1564 + 13 + 24 + + + 2965 + 1576 + + + + + + + + Angle in degrees + 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+ + + + + + + d1a28e95-cf96-4936-bf34-8bf142d731bf + Construct Domain + + + + + Create a numeric domain from two numeric extremes. + 14534673-98f6-45e8-b11f-39da628bc4ca + Construct Domain + Dom + + + + + + 2193 + 764 + 61 + 44 + + + 2224 + 786 + + + + + + Start value of numeric domain + c60598bb-4ac8-4310-9f9f-0d75cea21cfe + Domain start + A + false + 64b83d9b-5ae1-4022-b79e-c9135d3cdfc6 + 1 + + + + + + 2195 + 766 + 14 + 20 + + + 2203.5 + 776 + + + + + + 1 + + + + + 1 + {0} + + + + + 0 + + + + + + + + + + + End value of numeric domain + c4f2cdcf-3ac3-420a-925f-c2653c042270 + Domain end + B + false + 2398bd00-b514-4723-b31f-c436e1ae908b + 1 + + + + + + 2195 + 786 + 14 + 20 + + + 2203.5 + 796 + + + + + + 1 + + + + + 1 + {0} + + + + + 1 + + + + + + + + + + + Numeric domain between {A} and {B} + 035d494c-d9fa-434d-874c-206716be82af + Domain + I + false + 0 + + + + + + 2239 + 766 + 13 + 40 + + + 2245.5 + 786 + + + + + + + + + + + + 9445ca40-cc73-4861-a455-146308676855 + Range + + + + + Create a range of numbers. + fbb4ab09-bb2a-42d6-a9a9-345b5f2473bc + Range + Range + + + + + + 2295 + 775 + 64 + 44 + + + 2326 + 797 + + + + + + Domain of numeric range + 0dd6e136-7641-43ed-94ff-d87c8b3a4a7f + Domain + D + false + 035d494c-d9fa-434d-874c-206716be82af + 1 + + + + + + 2297 + 777 + 14 + 20 + + + 2305.5 + 787 + + + + + + 1 + + + + + 1 + {0} + + + + + + 0 + 1 + + + + + + + + + + + + Number of steps + b991e2eb-aa24-4a37-b83b-4de89b217793 + Steps + N + false + cc8db470-d2c6-4bbc-92b0-9305165143f7 + 1 + + + + + + 2297 + 797 + 14 + 20 + + + 2305.5 + 807 + + + + + + 1 + + + + + 1 + {0} + + + + + 10 + + + + + + + + + + + 1 + Range of numbers + 9b2fc3d6-5abb-46f8-a49e-0ae9e8dc1647 + Range + R + false + 0 + + + + + + 2341 + 777 + 16 + 40 + + + 2349 + 797 + + + + + + + + + + + + 57da07bd-ecab-415d-9d86-af36d7073abc + Number Slider + + + + + Numeric slider for single values + cc8db470-d2c6-4bbc-92b0-9305165143f7 + Number Slider + num curves + false + 0 + + + + + + 1969 + 831 + 284 + 20 + + + 1969.186 + 831.0843 + + + + + + 0 + 1 + 0 + 400 + 1 + 0 + 200 + + + + + + + + + 079bd9bd-54a0-41d4-98af-db999015f63d + VB Script + + + + + Private Function IsSet(ByVal param As String) As Boolean ' Check if an input parameter has data + Dim i As Integer = Component.Params.IndexOfInputParam(param) + If i > -1 Then + Return Component.Params.Input.ElementAt(i).DataType > 1 ' input parameter DataType of 1 means it's not receiving input (internal or external) + Else + Msg("error", "Input parameter '" & param & "' not found") + Return False + End If + End Function + + Private Sub Msg(ByVal type As String, ByVal msg As String) ' Output an error, warning, or informational message + Select Case type + Case "error" + Component.AddRuntimeMessage(GH_RuntimeMessageLevel.Error, msg) + Print("Error: " & msg) + Case "warning" + Component.AddRuntimeMessage(GH_RuntimeMessageLevel.Warning, msg) + Print("Warning: " & msg) + Case "info" + Component.AddRuntimeMessage(GH_RuntimeMessageLevel.Remark, msg) + Print(msg) + End Select + End Sub + + ' Solve for the m parameter from length and width (reference {1} equation (34), except b = width and K(k) and E(k) should be K(m) and E(m)) + Private Function SolveMFromLenWid(ByVal L As Double, ByVal w As Double) As Double + If w = 0 Then + Return Defined.M_ZERO_W ' for the boundry condition width = 0, bypass the function and return the known m value + End If + + Dim n As Integer = 1 ' Iteration counter (quit if >MAXIT) + Dim lower As Double = 0 ' m must be within this range + Dim upper As Double = 1 + Dim m As Double + Dim cwl As Double + + Do While (upper - lower) > Defined.MAXERR AndAlso (n) < Defined.MAXIT ' Repeat until range narrow enough or MAXIT + m = (upper + lower) / 2 + cwl = 2 * EllipticE(m) / EllipticK(m) - 1 ' calculate w/L with the test value of m + If cwl < w / L Then ' compares the calculated w/L with the actual w/L then narrows the range of possible m + upper = m + Else + lower = m + End If + n += 1 + Loop + Return m + End Function + + ' Solve for the m parameter from length and height (reference {1} equation (33), except K(k) should be K(m) and k = sqrt(m)) + ' Note that it's actually possible to find 2 valid values for m (hence 2 width values) at certain height values + Private Function SolveMFromLenHt(ByVal L As Double, ByVal h As Double) As List(Of Double) + Dim n As Integer = 1 ' Iteration counter (quit if >MAXIT) + Dim lower As Double = 0 ' m must be within this range + Dim upper As Double = 1 + Dim twoWidths As Boolean = h / L >= Defined.DOUBLE_W_HL_RATIO And h / L < Defined.MAX_HL_RATIO ' check to see if h/L is within the range where 2 solutions for the width are possible + Dim m As Double + Dim mult_m As New List(Of Double) + Dim chl As Double + + If twoWidths Then + ' find the first of two possible solutions for m with the following limits: + lower = Defined.M_DOUBLE_W ' see constants at bottom of script + upper = Defined.M_MAXHEIGHT ' see constants at bottom of script + Do While (upper - lower) > Defined.MAXERR AndAlso (n) < Defined.MAXIT ' Repeat until range narrow enough or MAXIT + m = (upper + lower) / 2 + chl = Math.Sqrt(m) / EllipticK(m) ' calculate h/L with the test value of m + If chl > h / L Then ' compares the calculated h/L with the actual h/L then narrows the range of possible m + upper = m + Else + lower = m + End If + n += 1 + Loop + mult_m.Add(m) + + ' then find the second of two possible solutions for m with the following limits: + lower = Defined.M_MAXHEIGHT ' see constants at bottom of script + upper = 1 + Do While (upper - lower) > Defined.MAXERR AndAlso (n) < Defined.MAXIT ' Repeat until range narrow enough or MAXIT + m = (upper + lower) / 2 + chl = Math.Sqrt(m) / EllipticK(m) ' calculate h/L with the test value of m + If chl < h / L Then ' compares the calculated h/L with the actual h/L then narrows the range of possible m + upper = m + Else + lower = m + End If + n += 1 + Loop + + If m <= Defined.M_MAX Then ' return this m parameter only if it falls within the maximum useful value (above which the curve breaks down) + mult_m.Add(m) + End If + + Else + ' find the one possible solution for the m parameter + upper = Defined.M_DOUBLE_W ' limit the upper end of the search to the maximum value of m for which only one solution exists + Do While (upper - lower) > Defined.MAXERR AndAlso (n) < Defined.MAXIT ' Repeat until range narrow enough or MAXIT + m = (upper + lower) / 2 + chl = Math.Sqrt(m) / EllipticK(m) ' calculate h/L with the test value of m + If chl > h / L Then ' compares the calculated h/L with the actual h/L then narrows the range of possible m + upper = m + Else + lower = m + End If + n += 1 + Loop + mult_m.Add(m) + End If + + Return mult_m + End Function + + ' Solve for the m parameter from width and height (derived from reference {1} equations (33) and (34) with same notes as above) + Private Function SolveMFromWidHt(ByVal w As Double, ByVal h As Double) As Double + Dim n As Integer = 1 ' Iteration counter (quit if >MAXIT) + Dim lower As Double = 0 ' m must be within this range + Dim upper As Double = 1 + Dim m As Double + Dim cwh As Double + + Do While (upper - lower) > Defined.MAXERR AndAlso (n) < Defined.MAXIT ' Repeat until range narrow enough or MAXIT + m = (upper + lower) / 2 + cwh = (2 * EllipticE(m) - EllipticK(m)) / Math.Sqrt(m) ' calculate w/h with the test value of m + If cwh < w / h Then ' compares the calculated w/h with the actual w/h then narrows the range of possible m + upper = m + Else + lower = m + End If + n += 1 + Loop + + Return m + End Function + + ' Calculate length based on height and an m parameter, derived from reference {1} equation (33), except K(k) should be K(m) and k = sqrt(m) + Private Function Cal_L(ByVal h As Double, ByVal m As Double) As Double + Return h * EllipticK(m) / Math.Sqrt(m) + End Function + + ' Calculate width based on length and an m parameter, derived from reference {1} equation (34), except b = width and K(k) and E(k) should be K(m) and E(m) + Private Function Cal_W(ByVal L As Double, ByVal m As Double) As Double + Return L * (2 * EllipticE(m) / EllipticK(m) - 1) + End Function + + ' Calculate height based on length and an m parameter, from reference {1} equation (33), except K(k) should be K(m) and k = sqrt(m) + Private Function Cal_H(ByVal L As Double, ByVal m As Double) As Double + Return L * Math.Sqrt(m) / EllipticK(m) + End Function + + ' Calculate the unique m parameter based on a start tangent angle, from reference {2}, just above equation (9a), that states k = Sin(angle / 2 + Pi / 4), + ' but as m = k^2 and due to this script's need for an angle rotated 90Β° versus the one in reference {1}, the following formula is the result + ' New note: verified by reference {4}, pg. 78 at the bottom + Private Function Cal_M(ByVal a As Double) As Double + Return (1 - Math.Cos(a)) / 2 ' equal to Sin^2(a/2) too + End Function + + ' Calculate start tangent angle based on an m parameter, derived from above formula + Private Function Cal_A(ByVal m As Double) As Double + Return Math.Acos(1 - 2 * m) + End Function + + ' This is the heart of this script, taking the found (or specified) length, width, and angle values along with the found m parameter to create + ' a list of points that approximate the shape or form of the elastica. It works by finding the x and y coordinates (which are reversed versus + ' the original equations (12a) and (12b) from reference {2} due to the 90Β° difference in orientation) based on the tangent angle along the curve. + ' See reference {2} for more details on how they derived it. Note that to simplify things, the algorithm only calculates the points for half of the + ' curve, then mirrors those points along the y-axis. + Private Function FindBendForm(ByVal L As Double, ByVal w As Double, ByVal m As Double, ByVal ang As Double, ByVal refPln As Plane) As List(Of Point3d) + L = L / 2 ' because the below algorithm is based on the formulas in reference {2} for only half of the curve + w = w / 2 ' same + + If ang = 0 Then ' if angle (and height) = 0, then simply return the start and end points of the straight line + Dim out As New List(Of Point3d) + out.Add(refPln.PointAt(w, 0, 0)) + out.Add(refPln.PointAt(-w, 0, 0)) + Return out + End If + + Dim x As Double + Dim y As Double + Dim halfCurvePts As New List(Of Point3d) + Dim fullCurvePts As New List(Of Point3d) + Dim translatedPts As New List(Of Point3d) + + ang -= Math.PI / 2 ' a hack to allow this algorithm to work, since the original curve in paper {2} was rotated 90Β° + Dim angB As Double = ang + (-Math.PI / 2 - ang) / Defined.CURVEDIVS ' angB is the 'lowercase theta' which should be in formula {2}(12b) as the interval + ' start [a typo...see equation(3)]. It's necessary to start angB at ang + [interval] instead of just ang due to integration failing at angB = ang + halfCurvePts.Add(New Point3d(w, 0, 0)) ' start with this known initial point, as integration will fail when angB = ang + + ' each point {x, y} is calculated from the tangent angle, angB, that occurs at each point (which is why this iterates from ~ang to -pi/2, the known end condition) + Do While Math.Round(angB, Defined.ROUNDTO) >= Math.Round(-Math.PI / 2, Defined.ROUNDTO) + y = (Math.Sqrt(2) * Math.Sqrt(Math.Sin(ang) - Math.Sin(angB)) * (w + L)) / (2 * EllipticE(m)) ' note that x and y are swapped vs. (12a) and (12b) + x = (L / (Math.Sqrt(2) * EllipticK(m))) * Simpson(angB, -Math.PI / 2, 500, ang) ' calculate the Simpson approximation of the integral (function f below) + ' over the interval angB ('lowercase theta') to -pi/2. side note: is 500 too few iterations for the Simson algorithm? + + If Math.Round(x, Defined.ROUNDTO) = 0 Then x = 0 + halfCurvePts.Add(New Point3d(x, y, 0)) + + angB += (-Math.PI / 2 - ang) / Defined.CURVEDIVS ' onto the next tangent angle + Loop + + ' After finding the x and y values for half of the curve, add the {-x, y} values for the rest of the curve + For Each point As Point3d In halfCurvePts + If Math.Round(point.X, Defined.ROUNDTO) = 0 Then + If Math.Round(point.Y, Defined.ROUNDTO) = 0 Then + fullCurvePts.Add(New Point3d(0, 0, 0)) ' special case when width = 0: when x = 0, only duplicate the point when y = 0 too + End If + Else + fullCurvePts.Add(New Point3d(-point.X, point.Y, 0)) + End If + Next + halfCurvePts.Reverse + fullCurvePts.AddRange(halfCurvePts) + + For Each p As Point3d In fullCurvePts + translatedPts.Add(refPln.PointAt(p.X, p.Y, p.Z)) ' translate the points from the reference plane to the world plane + Next + + Return translatedPts + End Function + + ' Interpolates the points from FindBendForm to create the Elastica curve. Uses start & end tangents for greater accuracy. + Private Function MakeCurve(ByVal pts As List(Of Point3d), ByVal ang As Double, ByVal refPln As Plane) As Curve + If ang <> 0 Then + Dim ts, te As New Vector3d(refPln.XAxis) + ts.Rotate(ang, refPln.ZAxis) + te.Rotate(-ang, refPln.ZAxis) + Return Curve.CreateInterpolatedCurve(pts, 3, CurveKnotStyle.Chord, ts, te) ' 3rd degree curve with 'Chord' Knot Style + Else + Return Curve.CreateInterpolatedCurve(pts, 3) ' if angle (and height) = 0, then simply interpolate the straight line (no start/end tangents) + End If + End Function + + ' Implements the Simpson approximation for an integral of function f below + Public Function Simpson(a As Double, b As Double, n As Integer, theta As Double) As Double 'n should be an even number + Dim j As Integer, s1 As Double, s2 As Double, h As Double + h = (b - a) / n + s1 = 0 + s2 = 0 + For j = 1 To n - 1 Step 2 + s1 = s1 + fn(a + j * h, theta) + Next j + For j = 2 To n - 2 Step 2 + s2 = s2 + fn(a + j * h, theta) + Next j + Simpson = h / 3 * (fn(a, theta) + 4 * s1 + 2 * s2 + fn(b, theta)) + End Function + + ' Specific calculation for the above integration + Public Function fn(x As Double, theta As Double) As Double + fn = Math.Sin(x) / (Math.Sqrt(Math.Sin(theta) - Math.Sin(x))) ' from reference {2} formula (12b) + End Function + + + ' Return the Complete Elliptic integral of the 1st kind + ' Abramowitz and Stegun p.591, formula 17.3.11 + ' Code from http://www.codeproject.com/Articles/566614/Elliptic-integrals + Public Function EllipticK(ByVal m As Double) As Double + Dim sum, term, above, below As Double + sum = 1 + term = 1 + above = 1 + below = 2 + + For i As Integer = 1 To 100 + term *= above / below + sum += Math.Pow(m, i) * Math.Pow(term, 2) + above += 2 + below += 2 + Next + sum *= 0.5 * Math.PI + Return sum + End Function + + + ' Return the Complete Elliptic integral of the 2nd kind + ' Abramowitz and Stegun p.591, formula 17.3.12 + ' Code from http://www.codeproject.com/Articles/566614/Elliptic-integrals + Public Function EllipticE(ByVal m As Double) As Double + Dim sum, term, above, below As Double + sum = 1 + term = 1 + above = 1 + below = 2 + + For i As Integer = 1 To 100 + term *= above / below + sum -= Math.Pow(m, i) * Math.Pow(term, 2) / above + above += 2 + below += 2 + Next + sum *= 0.5 * Math.PI + Return sum + End Function + + Friend Partial NotInheritable Class Defined + Private Sub New() + End Sub + + ' Note: most of these values for m and h/L ratio were found with Wolfram Alpha and either specific intercepts (x=0) or local minima/maxima. They should be constant. + Public Const M_SKETCHY As Double = 0.95 ' value of the m parameter where the curvature near the ends of the curve gets wonky + Public Const M_MAX As Double = 0.993 ' maximum useful value of the m parameter, above which this algorithm for the form of the curve breaks down + Public Const M_ZERO_W As Double = 0.826114765984970336 ' value of the m parameter when width = 0 + Public Const M_MAXHEIGHT As Double = 0.701327460663101223 ' value of the m parameter at maximum possible height of the bent rod/wire + Public Const M_DOUBLE_W As Double = 0.180254422335013983 ' minimum value of the m parameter when two width values are possible for a given height and length + Public Const DOUBLE_W_HL_RATIO As Double = 0.257342117984635757 ' value of the height/length ratio above which there are two possible width values + Public Const MAX_HL_RATIO As Double = 0.403140189705650243 ' maximum possible value of the height/length ratio + + Public Const MAXERR As Double = 0.0000000001 ' error tolerance + Public Const MAXIT As Integer = 100 ' maximum number of iterations + Public Const ROUNDTO As Integer = 10 ' number of decimal places to round off to + Public Const CURVEDIVS As Integer = 50 ' number of sample points for building the curve (or half-curve as it were) + End Class + A VB.NET scriptable component + + 98 + 86 + + true + 63218c71-7758-4168-b163-745e456fb525 + VB Script + VB + true + 0 + ' ----------------------------------------------------------------- + ' Elastic Bending Script by Will McElwain + ' Created February 2014 + ' + ' DESCRIPTION: + ' This beast creates the so-called 'elastica curve', the shape a long, thin rod or wire makes when it is bent elastically (i.e. not permanently). In this case, force + ' is assumed to only be applied horizontally (which would be in line with the rod at rest) and both ends are assumed to be pinned or hinged meaning they are free + ' to rotate (as opposed to clamped, when the end tangent angle is fixed, usually horizontally). An interesting finding is that it doesn't matter what the material or + ' cross-sectional area is, as long as they're uniform along the entire length. Everything makes the same shape when bent as long as it doesn't cross the threshold + ' from elastic to plastic (permanent) deformation (I don't bother to find that limit here, but can be found if the yield stress for a material is known). + ' + ' Key to the formulas used in this script are elliptic integrals, specifically K(m), the complete elliptic integral of the first kind, and E(m), the complete elliptic + ' integral of the second kind. There was a lot of confusion over the 'm' and 'k' parameters for these functions, as some people use them interchangeably, but they are + ' not the same. m = k^2 (thus k = Sqrt(m)). I try to use the 'm' parameter exclusively to avoid this confusion. Note that there is a unique 'm' parameter for every + ' configuration/shape of the elastica curve. + ' + ' This script tries to find that unique 'm' parameter based on the inputs. The algorithm starts with a test version of m, evaluates an expression, say 2*E(m)/K(m)-1, + ' then compares the result to what it should be (in this case, a known width/length ratio). Iterate until the correct m is found. Once we have m, we can then calculate + ' all of the other unknowns, then find points that lie on that curve, then interpolate those points for the actual curve. You can also use Wolfram|Alpha as I did to + ' find the m parameter based on the equations in this script (example here: http://tiny.cc/t4tpbx for when say width=45.2 and length=67.1). + ' + ' Other notes: + ' * This script works with negative values for width, which will creat a self-intersecting curve (as it should). The curvature of the elastica starts to break down around + ' m=0.95 (~154Β°), but this script will continue to work until M_MAX, m=0.993 (~169Β°). If you wish to ignore self-intersecting curves, set ignoreSelfIntersecting to True + ' * When the only known values are length and height, it is actually possible for certain ratios of height to length to have two valid m values (thus 2 possible widths + ' and angles). This script will return them both. + ' * Only the first two valid parameters (of the required ones) will be used, meaning if all four are connected (length, width or a PtB, height, and angle), this script will + ' only use length and width (or a PtB). + ' * Depending on the magnitude of your inputs (say if they're really small, like if length < 10), you might have to increase the constant ROUNDTO at the bottom + ' + ' REFERENCES: + ' {1} "The elastic rod" by M.E. Pacheco Q. & E. Pina, http://www.scielo.org.mx/pdf/rmfe/v53n2/v53n2a8.pdf + ' {2} "An experiment in nonlinear beam theory" by A. Valiente, http://www.deepdyve.com/lp/doc/I3lwnxdfGz , also here: http://tiny.cc/Valiente_AEiNBT + ' {3} "Snap buckling, writhing and Loop formation In twisted rods" by V.G.A. GOSS, http://myweb.lsbu.ac.uk/~gossga/thesisFinal.pdf + ' {4} "Theory of Elastic Stability" by Stephen Timoshenko, http://www.scribd.com/doc/50402462/Timoshenko-Theory-of-Elastic-Stability (start on p. 76) + ' + ' INPUT: + ' PtA - First anchor point (required) + ' PtB - Second anchor point (optional, though 2 out of the 4--length, width, height, angle--need to be specified) + ' [note that PtB can be the same as PtA (meaning width would be zero)] + ' [also note that if a different width is additionally specified that's not equal to the distance between PtA and PtB, then the end point will not equal PtB anymore] + ' Pln - Plane of the bent rod/wire, which bends up in the +y direction. The line between PtA and PtB (if specified) must be parallel to the x-axis of this plane + ' + ' ** 2 of the following 4 need to be specified ** + ' Len - Length of the rod/wire, which needs to be > 0 + ' Wid - Width between the endpoints of the curve [note: if PtB is specified in addition, and distance between PtA and PtB <> width, the end point will be relocated + ' Ht - Height of the bent rod/wire (when negative, curve will bend downward, relative to the input plane, instead) + ' Ang - Inner departure angle or tangent angle (in radians) at the ends of the bent rod/wire. Set up so as width approaches length (thus height approaches zero), angle approaches zero + ' + ' * Following variables only needed for optional calculating of bending force, not for shape of curve. + ' E - Young's modulus (modulus of elasticity) in GPa (=N/m^2) (material-specific. for example, 7075 aluminum is roughly 71.7 GPa) + ' I - Second moment of area (or area moment of inertia) in m^4 (cross-section-specific. for example, a hollow rod + ' would have I = pi * (outer_diameter^4 - inner_diameter^4) / 32 + ' Note: E*I is also known as flexural rigidity or bending stiffness + ' + ' OUTPUT: + ' out - only for debugging messages + ' Pts - the list of points that approximate the shape of the elastica + ' Crv - the 3rd-degree curve interpolated from those points (with accurate start & end tangents) + ' L - the length of the rod/wire + ' W - the distance (width) between the endpoints of the rod/wire + ' H - the height of the bent rod/wire + ' A - the tangent angle at the (start) end of the rod/wire + ' F - the force needed to hold the rod/wire in a specific shape (based on the material properties & cross-section) **be sure your units for 'I' match your units for the + ' rest of your inputs (length, width, etc.). Also note that the critical buckling load (force) that makes the rod/wire start to bend can be found at height=0 + ' + ' THANKS TO: + ' MΓ₯rten Nettelbladt (thegeometryofbending.blogspot.com) + ' Daniel Piker (Kangaroo plugin) + ' David Rutten (Grasshopper guru) + ' Euler & Bernoulli (the O.G.'s) + ' + ' ----------------------------------------------------------------- + + Dim ignoreSelfIntersecting As Boolean = False ' set to True if you don't want to output curves where width < 0, which creates a self-intersecting curve + + Dim inCt As Integer = 0 ' count the number of required parameters that are receiving data + Dim length As Double + Dim width As System.Object = Nothing ' need to set as Nothing so we can check if it has been assigned a value later + Dim height As Double + Dim angle As Double + Dim m As Double + Dim multiple_m As New List(Of Double) + Dim AtoB As Line + Dim flip_H As Boolean = False ' if height is negative, this flag will be set + Dim flip_A As Boolean = False ' if angle is negative, this flag will be set + + If Not IsSet("Pln") Then + Msg("error", "Base plane is not set") + Return + End If + + If Not IsSet("PtA") Then + Msg("error", "Point A is not set") + Return + End If + + If Math.Round(Pln.DistanceTo(PtA), Defined.ROUNDTO) <> 0 Then + Msg("error", "Point A is not on the base plane") + Return + End If + + Dim refPlane As Plane = Pln ' create a reference plane = input plane and set the origin of it to PtA in case PtA isn't the origin already + refPlane.Origin = PtA + + If IsSet("PtB") Then + If Math.Round(Pln.DistanceTo(PtB), Defined.ROUNDTO) <> 0 Then + Msg("error", "Point B is not on the base plane") + Return + End If + + AtoB = New Line(PtA, PtB) + If AtoB.Length <> 0 And Not AtoB.Direction.IsPerpendicularTo(Pln.YAxis) Then + Msg("error", "The line between PtA and PtB is not perpendicular to the Y-axis of the specified plane") + Return + End If + + inCt += 1 + If IsSet("Wid") Then Msg("info", "Wid will override the distance between PtA and PtB. If you do not want this to happen, disconnect PtB or Wid.") + + width = PtA.DistanceTo(PtB) ' get the width (distance) between PtA and PtB + + Dim refPtB As Point3d + refPlane.RemapToPlaneSpace(PtB, refPtB) + If refPtB.X < 0 Then width = -width ' check if PtB is to the left of PtA...if so, width is negative + End If + + If IsSet("Len") Then inCt += 1 + If IsSet("Wid") Then inCt += 1 + If IsSet("Ht") Then inCt += 1 + If IsSet("Ang") Then inCt += 1 + If inCt > 2 Then Msg("info", "More parameters set than are required (out of length, width, height, angle). Only using the first two valid ones.") + + ' check for connected/specified inputs. note: only the first two that it comes across will be used + If IsSet("Len") Then ' if length is specified then... + If Len <= 0 Then + Msg("error", "Length cannot be negative or zero") + Return + End If + If IsSet("Wid") Then ' find height & angle based on length and specified width + If Wid > Len Then + Msg("error", "Width is greater than length") + Return + End If + If Wid = Len Then ' skip the solver and set the known values + height = 0 + m = 0 + angle = 0 + width = Wid + Else + m = SolveMFromLenWid(Len, Wid) + height = Cal_H(Len, m) ' L * Sqrt(m) / K(m) + angle = Cal_A(m) ' Acos(1 - 2 * m) + width = Wid + End If + + Else If width IsNot Nothing Then ' find height & angle based on length and calculated width (distance between PtA and PtB) + If width > Len Then + Msg("error", "Width is greater than length") + Return + End If + If width = Len Then ' skip the solver and set the known values + height = 0 + m = 0 + angle = 0 + Else + m = SolveMFromLenWid(Len, width) + height = Cal_H(Len, m) ' L * Sqrt(m) / K(m) + angle = Cal_A(m) ' Acos(1 - 2 * m) + End If + + Else If IsSet("Ht") Then ' find width & angle based on length and height ** possible to return 2 results ** + If Math.Abs(Ht / Len) > Defined.MAX_HL_RATIO Then + Msg("error", "Height not possible with given length") + Return + End If + If Ht < 0 Then + Ht = -Ht ' if height is negative, set it to positive (for the calculations) but flip the reference plane about its x-axis + refPlane.Transform(Transform.Mirror(New Plane(refPlane.Origin, refPlane.XAxis, refPlane.ZAxis))) + flip_A = True + flip_H = True + End If + If Ht = 0 Then ' skip the solver and set the known values + width = Len + angle = 0 + Else + multiple_m = SolveMFromLenHt(Len, Ht) ' note that it's possible for two values of m to be found if height is close to max height + If multiple_m.Count = 1 Then ' if there's only one m value returned, calculate the width & angle here. we'll deal with multiple m values later + m = multiple_m.Item(0) + width = Cal_W(Len, m) ' L * (2 * E(m) / K(m) - 1) + angle = Cal_A(m) ' Acos(1 - 2 * m) + End If + End If + height = Ht + + Else If IsSet("Ang") Then ' find width & height based on length and angle + If Ang < 0 Then + Ang = -Ang ' if angle is negative, set it to positive (for the calculations) but flip the reference plane about its x-axis + refPlane.Transform(Transform.Mirror(New Plane(refPlane.Origin, refPlane.XAxis, refPlane.ZAxis))) + flip_A = True + flip_H = True + End If + m = Cal_M(Ang) ' (1 - Cos(a)) / 2 + If Ang = 0 Then ' skip the solver and set the known values + width = Len + height = 0 + Else + width = Cal_W(Len, m) ' L * (2 * E(m) / K(m) - 1) + height = Cal_H(Len, m) ' L * Sqrt(m) / K(m) + End If + angle = Ang + + Else + Msg("error", "Need to specify one more parameter in addition to length") + Return + End If + length = Len + + Else If IsSet("Wid") Then ' if width is specified then... + If IsSet("Ht") Then ' find length & angle based on specified width and height + If Ht < 0 Then + Ht = -Ht ' if height is negative, set it to positive (for the calculations) but flip the reference plane about its x-axis + refPlane.Transform(Transform.Mirror(New Plane(refPlane.Origin, refPlane.XAxis, refPlane.ZAxis))) + flip_A = True + flip_H = True + End If + If Ht = 0 Then ' skip the solver and set the known values + length = Wid + angle = 0 + Else + m = SolveMFromWidHt(Wid, Ht) + length = Cal_L(Ht, m) ' h * K(m) / Sqrt(m) + angle = Cal_A(m) ' Acos(1 - 2 * m) + End If + height = Ht + + Else If IsSet("Ang") Then ' find length & height based on specified width and angle + If Wid = 0 Then + Msg("error", "Curve not possible with width = 0 and an angle as inputs") + Return + End If + If Ang < 0 Then + Ang = -Ang ' if angle is negative, set it to positive (for the calculations) but flip the reference plane about its x-axis + refPlane.Transform(Transform.Mirror(New Plane(refPlane.Origin, refPlane.XAxis, refPlane.ZAxis))) + flip_A = True + flip_H = True + End If + m = Cal_M(Ang) ' (1 - Cos(a)) / 2 + If Ang = 0 Then ' skip the solver and set the known values + length = Wid + height = 0 + Else + length = Wid / (2 * EllipticE(m) / EllipticK(m) - 1) + If length < 0 Then + Msg("error", "Curve not possible at specified width and angle (calculated length is negative)") + Return + End If + height = Cal_H(length, m) ' L * Sqrt(m) / K(m) + End If + angle = Ang + + Else + Msg("error", "Need to specify one more parameter in addition to width (Wid)") + Return + End If + width = Wid + + Else If width IsNot Nothing Then ' if width is determined by PtA and PtB then... + If IsSet("Ht") Then ' find length & angle based on calculated width and height + If Ht < 0 Then + Ht = -Ht ' if height is negative, set it to positive (for the calculations) but flip the reference plane about its x-axis + refPlane.Transform(Transform.Mirror(New Plane(refPlane.Origin, refPlane.XAxis, refPlane.ZAxis))) + flip_A = True + flip_H = True + End If + If Ht = 0 Then ' skip the solver and set the known values + length = width + angle = 0 + Else + m = SolveMFromWidHt(width, Ht) + length = Cal_L(Ht, m) ' h * K(m) / Sqrt(m) + angle = Cal_A(m) ' Acos(1 - 2 * m) + End If + height = Ht + + Else If IsSet("Ang") Then ' find length & height based on calculated width and angle + If width = 0 Then + Msg("error", "Curve not possible with width = 0 and an angle as inputs") + Return + End If + If Ang < 0 Then + Ang = -Ang ' if angle is negative, set it to positive (for the calculations) but flip the reference plane about its x-axis + refPlane.Transform(Transform.Mirror(New Plane(refPlane.Origin, refPlane.XAxis, refPlane.ZAxis))) + flip_A = True + flip_H = True + End If + m = Cal_M(Ang) ' (1 - Cos(a)) / 2 + If Ang = 0 Then ' skip the solver and set the known values + length = width + height = 0 + Else + length = width / (2 * EllipticE(m) / EllipticK(m) - 1) + If length < 0 Then + Msg("error", "Curve not possible at specified width and angle (calculated length is negative)") + Return + End If + height = Cal_H(length, m) ' L * Sqrt(m) / K(m) + End If + angle = Ang + + Else + Msg("error", "Need to specify one more parameter in addition to PtA and PtB") + Return + End If + + Else If IsSet("Ht") Then ' if height is specified then... + If IsSet("Ang") Then ' find length & width based on height and angle + If Ht < 0 Then + Ht = -Ht ' if height is negative, set it to positive (for the calculations) but flip the reference plane about its x-axis + refPlane.Transform(Transform.Mirror(New Plane(refPlane.Origin, refPlane.XAxis, refPlane.ZAxis))) + flip_H = True + flip_A = True + End If + If Ht = 0 Then + Msg("error", "Height can't = 0 if only height and angle are specified") + Return + Else + If Ang < 0 Then + Ang = -Ang ' if angle is negative, set it to positive (for the calculations) but flip the reference plane about its x-axis + refPlane.Transform(Transform.Mirror(New Plane(refPlane.Origin, refPlane.XAxis, refPlane.ZAxis))) + flip_A = Not flip_A + flip_H = Not flip_H + End If + m = Cal_M(Ang) ' (1 - Cos(a)) / 2 + If Ang = 0 Then + Msg("error", "Angle can't = 0 if only height and angle are specified") + Return + Else + length = Cal_L(Ht, m) ' h * K(m) / Sqrt(m) + width = Cal_W(length, m) ' L * (2 * E(m) / K(m) - 1) + End If + angle = Ang + End If + height = Ht + + Else + Msg("error", "Need to specify one more parameter in addition to height") + Return + End If + + Else If IsSet("Ang") Then + Msg("error", "Need to specify one more parameter in addition to angle") + Return + Else + Msg("error", "Need to specify two of the four parameters: length, width (or PtB), height, and angle") + Return + End If + + If m > Defined.M_MAX Then + Msg("error", "Form of curve not solvable with current algorithm and given inputs") + Return + End If + + refPlane.Origin = refPlane.PointAt(width / 2, 0, 0) ' adjust the origin of the reference plane so that the curve is centered about the y-axis (start of the curve is at x = -width/2) + + If multiple_m.Count > 1 Then ' if there is more than one m value returned, calculate the width, angle, and curve for each + Dim multi_pts As New DataTree(Of Point3d) + Dim multi_crv As New List(Of Curve) + Dim tmp_pts As New List(Of Point3d) + Dim multi_W, multi_A, multi_F As New List(Of Double) + Dim j As Integer = 0 ' used for creating a new branch (GH_Path) for storing pts which is itself a list of points + + For Each m_val As Double In multiple_m + width = Cal_W(length, m_val) 'length * (2 * EllipticE(m_val) / EllipticK(m_val) - 1) + + If width < 0 And ignoreSelfIntersecting Then + Msg("warning", "One curve is self-intersecting. To enable these, set ignoreSelfIntersecting to False") + Continue For + End If + + If m_val >= Defined.M_SKETCHY Then Msg("info", "Accuracy of the curve whose width = " & Math.Round(width, 4) & " is not guaranteed") + + angle = Cal_A(m_val) 'Math.Asin(2 * m_val - 1) + refPlane.Origin = refPlane.PointAt(width / 2, 0, 0) ' adjust the origin of the reference plane so that the curve is centered about the y-axis (start of the curve is at x = -width/2) + + tmp_pts = FindBendForm(length, width, m_val, angle, refPlane) + multi_pts.AddRange(tmp_pts, New GH_Path(j)) + multi_crv.Add(MakeCurve(tmp_pts, angle, refPlane)) + + multi_W.Add(width) + If flip_A Then angle = -angle + multi_A.Add(angle) + + E = E * 10 ^ 9 ' Young's modulus input E is in GPa, so we convert to Pa here (= N/m^2) + multi_F.Add(EllipticK(m_val) ^ 2 * E * I / length ^ 2) ' from reference {4} pg. 79 + + j += 1 + refPlane.Origin = PtA ' reset the reference plane origin to PtA for the next m_val + 'Print("length=" & length & ", width=" & width & ", height=" & height & ", angle=" & angle & ", m=" & m_val & ", k=" & Math.Sqrt(m_val) & ", w/L=" & width / length & ", h/L=" & height / length & ", w/h=" & width / height) + Next + + ' assign the outputs + Pts = multi_pts + Crv = multi_crv + L = length + W = multi_W + If flip_H Then height = -height + H = height + A = multi_A + F = multi_F + + Else ' only deal with the single m value + If m >= Defined.M_SKETCHY Then Msg("info", "Accuracy of the curve at these parameters is not guaranteed") + + If width < 0 And ignoreSelfIntersecting Then + Msg("error", "Curve is self-intersecting. To enable these, set ignoreSelfIntersecting to False") + Return + End If + + Pts = FindBendForm(length, width, m, angle, refPlane) + Crv = MakeCurve(pts, angle, refPlane) + L = length + W = width + If flip_H Then height = -height + H = height + If flip_A Then angle = -angle + A = angle + + E = E * 10 ^ 9 ' Young's modulus input E is in GPa, so we convert to Pa here (= N/m^2) + F = EllipticK(m) ^ 2 * E * I / length ^ 2 ' from reference {4} pg. 79. Note: the critical buckling (that makes the rod/wire start to bend) can be found at height=0 (width=length) + + 'height = Math.Sqrt(((2 * Len / 5) ^ 2 - ((Wid - Len / 5) / 2) ^ 2) ' quick approximation discovered by MΓ₯rten of 'Geometry of Bending' fame ( http://tiny.cc/it2pbx ) + 'width = (Len +/- 2 * Math.Sqrt(4 * Len ^ 2 - 25 * Ht ^ 2)) / 5 ' derived from above + 'length = (2 * Math.Sqrt(15 * Ht ^ 2 + 4 * Wid ^ 2) - Wid) / 3 ' derived from above + + 'Print("length=" & length & ", width=" & width & ", height=" & height & ", angle=" & angle & ", m=" & m & ", k=" & Math.Sqrt(m) & ", w/L=" & width / length & ", h/L=" & height / length & ", w/h=" & width / height) + End If + + + + + + + 2397 + 703 + 84 + 184 + + + 2439 + 795 + + + + + + 9 + 84fa917c-1ed8-4db3-8be1-7bdc4a6495a2 + 84fa917c-1ed8-4db3-8be1-7bdc4a6495a2 + 84fa917c-1ed8-4db3-8be1-7bdc4a6495a2 + 84fa917c-1ed8-4db3-8be1-7bdc4a6495a2 + 84fa917c-1ed8-4db3-8be1-7bdc4a6495a2 + 84fa917c-1ed8-4db3-8be1-7bdc4a6495a2 + 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