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- <item name="AdditionalSource" type_name="gh_string" type_code="10"> 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 90° 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 90° 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 90°
- 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</item>
- <item name="Description" type_name="gh_string" type_code="10">A VB.NET scriptable component</item>
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- <item name="ScriptSource" type_name="gh_string" type_code="10"> ' -----------------------------------------------------------------
- ' 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 (~154°), but this script will continue to work until M_MAX, m=0.993 (~169°). 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:
- ' Mårten 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 Mårten 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
- </item>
- </items>
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- <items count="2">
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- <item name="AdditionalSource" type_name="gh_string" type_code="10"> 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 90° 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 90° 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 90°
- 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</item>
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- <item name="ScriptSource" type_name="gh_string" type_code="10"> ' -----------------------------------------------------------------
- ' 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 (~154°), but this script will continue to work until M_MAX, m=0.993 (~169°). 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:
- ' Mårten 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 Mårten 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
- </item>
- </items>
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- <!--GH_Point only stores data which exists, test the existence of data fields prior to reading.-->
- <items count="2">
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- <X>0</X>
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</bitmap>
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