(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.0' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 113364, 2809]*) (*NotebookOutlinePosition[ 114049, 2833]*) (* CellTagsIndexPosition[ 114005, 2829]*) (*WindowFrame->Normal*) Notebook[{ Cell[TextData[{ "\"Visualizing Quaternions\"\nAndrew J. Hanson\nIndiana University\n\ 2004-2005\n\nMaster Utility File for quatbook.tex ", StyleBox["Mathematica", FontSlant->"Italic"], " files" }], "Section"], Cell[CellGroupData[{ Cell["Utilities for plotting ", "Section"], Cell[BoxData[ \(<< Graphics`ParametricPlot3D`\)], "Input", InitializationCell->True], Cell[BoxData[ \(MakePolygons[vl_List] := Module[{l = vl, l1 = Map[RotateLeft, vl], meshxx}, meshxx = {l, l1, RotateLeft[l1], RotateLeft[l]}; \[IndentingNewLine]meshxx = Map[Drop[#, \(-1\)] &, meshxx, {1}]; \[IndentingNewLine]meshxx = Map[Drop[#, \(-1\)] &, meshxx, {2}]; \[IndentingNewLine]Polygon /@ Transpose[Map[Flatten[#, 1] &, meshxx]]]\)], "Input", Editable->False, InitializationCell->True], Cell[BoxData[ \(MakeWireFrame[vl_List] := Module[{l = vl, l1 = Map[RotateLeft, vl], meshxx, polys}, meshxx = {l, l1, RotateLeft[l1], RotateLeft[l]}; \[IndentingNewLine]meshxx = Map[Drop[#, \(-1\)] &, meshxx, {1}]; \[IndentingNewLine]meshxx = Map[Drop[#, \(-1\)] &, meshxx, {2}]; \[IndentingNewLine]polys\ = Polygon /@ Transpose[ Map[Flatten[#, 1] &, meshxx]]; \[IndentingNewLine]polys /. \((Polygon[ pts_] \[RuleDelayed] Line[Append[pts, First[pts]]])\)]\)], "Input", Editable->False, InitializationCell->True], Cell[BoxData[ \(wireframeParPlot3D[ eqn_, {u_, u0_, u1_, du_}, {v_, v0_, v1_, dv_}]\ := \[IndentingNewLine]Module[{plot, graphics}, \[IndentingNewLine]plot\ = \ ParametricPlot3D[eqn, {u, u0, u1, du}, {v, v0, v1, dv}, DisplayFunction \[Rule] Identity]; \[IndentingNewLine]graphics = plot /. \((Polygon[pts_] \[RuleDelayed] Line[Append[pts, First[pts]]])\); \[IndentingNewLine]Show[ graphics, DisplayFunction \[Rule] $DisplayFunction]]\)], "Input", Editable->False, InitializationCell->True], Cell[BoxData[ \(meshviewDOT[file_, ptlist_, ptsize_]\ := \[IndentingNewLine]Export[ file, \[IndentingNewLine]Join[{{"\"}, {ToString[ ptsize]}, {ToString[Length[ptlist]]}}, ptlist], \[IndentingNewLine]"\"]\)], "Input", Editable->False, InitializationCell->True], Cell[BoxData[ \(meshviewDOTC[file_, ptlist_, color4_, ptsize_]\ := \[IndentingNewLine]Export[ file, \[IndentingNewLine]Join[{{"\"}, color4, {ToString[ptsize]}, {ToString[Length[ptlist]]}}, ptlist], \[IndentingNewLine]"\"]\)], "Input", Editable->False, InitializationCell->True], Cell[BoxData[ \(sphere[nth_: 16, nph_: 16, r_: 1.0]\ := \ Module[{}, \n\t\t\ \ Table[{r\ Cos[th]\ Sin[ph], r\ Sin[th]\ Sin[ph], \ r\ Cos[ph]} // N, \n\t\t\t\ \ \ {th, 0, 2\ Pi, 2\ \ Pi/nth}, \ {ph, \(-\ Pi\), \ Pi, 2\ \ Pi/nph}]]\)], "Input", InitializationCell->True], Cell[BoxData[ \(wiresphere[\ nth_: 16, nph_: 16, r_: 1.0]\ := \ Graphics3D[ MakePolygons[ sphere[nth, nph, r]] /. \((Polygon[pts_]\ :> Line[Append[pts, First[pts]]])\)]\)], "Input", InitializationCell->True] }, Open ]], Cell[BoxData[ \(drawframe[frame3dmat_, xform_]\ := \[IndentingNewLine]Module[{lines}, \ \[IndentingNewLine]lines\ = \ Map[Line[{{0, 0, 0}, #}] &, Transpose[\((xform\ . \ frame3dmat)\)]]; \[IndentingNewLine]\ Show[Graphics3D[AbsoluteThickness[3]], Graphics3D[{RGBColor[1, 0, 0], lines[\([1]\)]}], \[IndentingNewLine]Graphics3D[{RGBColor[0, 1, 0], lines[\([2]\)]}], \[IndentingNewLine]Graphics3D[{RGBColor[ 0, 0, 1], lines[\([3]\)]}], \[IndentingNewLine]Axes \[Rule] True, \ AxesLabel \[Rule] {"\", \ "\", \ "\"}, \[IndentingNewLine]PlotRange \[Rule] {{\(-1\), 1}, {\(-1\), 1}, {\(-1\), 1}}]\ \[IndentingNewLine]]\)], "Input", InitializationCell->True], Cell[CellGroupData[{ Cell["\<\ Quaternion Library: q[[1]] --> q0 = Cos[th/2] q[[2,3,4]]] --> qvec = Sin[th/2] nhat\ \>", "Section"], Cell["\<\ qprod[q_List,p_List] := {q[[1]]*p[[1]] - q[[2]]*p[[2]] - q[[3]]*p[[3]] - q[[4]]*p[[4]], q[[1]]*p[[2]] + q[[2]]*p[[1]] + q[[3]]*p[[4]] - q[[4]]*p[[3]], q[[1]]*p[[3]] + q[[3]]*p[[1]] + q[[4]]*p[[2]] - q[[2]]*p[[4]], q[[1]]*p[[4]] + q[[4]]*p[[1]] + q[[2]]*p[[3]] - q[[3]]*p[[2]]} /; Length[q] == Length[p] == 4\ \>", "Input", Editable->False, InitializationCell->True, AspectRatioFixed->True], Cell["qinverse[q_List] := {q[[1]], - q[[2]], -q[[3]], - q[[4]]}", "Input", Editable->False, InitializationCell->True, AspectRatioFixed->True], Cell[BoxData[ \(\(\(normalize[ lst_List]\)\(\ \)\(:=\)\(\ \)\(Module[{norm\ = \ lst\ . \ lst, eps = 10^\((\(-14\))\)}, \[IndentingNewLine]If[ NumberQ[norm], \[IndentingNewLine]If[norm < eps, \ lst, \ N[lst/Sqrt[norm]]\ ], \[IndentingNewLine]lst/ Sqrt[norm]]]\)\(\ \ \)\)\)], "Input", Editable->False, InitializationCell->True], Cell["\<\ proj4 [pt4D_,angle_:0] := Chop[{pt4D[[1]],pt4D[[2]], Cos[angle]*pt4D[[3]] + Sin[angle]*pt4D[[4]]}]//N\ \>", \ "Input", Editable->False, InitializationCell->True, AspectRatioFixed->True], Cell["\<\ ROTATIONS and quaternions: q*v*qbar with v= (0,vec) a \"pure\" quaternion gives EACH COLUMN of a rotation matrix:\ \>", "Subsection"], Cell["\<\ QuatToRot[q_List] := Module[{q0= q[[1]], q1=q[[2]], q2 = q[[3]], q3 = q[[4]]}, Module[{d23 = 2 q2 q3, d1 = 2 q0 q1, d31 = 2 q3 q1, d2 = 2 q0 q2, d12 = 2 q1 q2, d3 = 2 q0 q3, q0sq = q0^2, q1sq = q1^2, q2sq = q2^2, q3sq = q3^2}, {{q0sq + q1sq - q2sq - q3sq, d12 - d3, d31 + d2}, {d12 + d3, q0sq - q1sq + q2sq - q3sq, d23 - d1}, {d31 - d2, d23 + d1, q0sq - q1sq - q2sq + q3sq}} ]] \ \>", "Input", Editable->False, InitializationCell->True, AspectRatioFixed->True], Cell["\<\ Warning: these are the reverse of conventional argument order: change order once all is consistent to (angle, nhat).\ \>", "Subsection"], Cell["\<\ makeQRot[n_List:{0,0,1},angle_:0] := Module[{c = Cos[angle/2], s = Sin[angle/2]}, {c, n[[1]]*s, n[[2]]*s, n[[3]]*s}//N]\ \>", "Input", Editable->False, InitializationCell->True, AspectRatioFixed->True], Cell["\<\ makeRotation[n_List:{0,0,1},angle_:0] := Module[{c = Cos[angle]//N, s = Sin[angle]//N, cm}, cm = 1 - c; {{c + cm*n[[1]]^2, cm*n[[2]]*n[[1]] - s*n[[3]], cm*n[[3]]*n[[1]] + s*n[[2]]}, {cm*n[[1]]*n[[2]] + s*n[[3]], c + cm*n[[2]]^2, cm*n[[3]]*n[[2]] - s*n[[1]]}, { cm*n[[1]]*n[[3]] - s*n[[2]], cm*n[[2]]*n[[3]] + s*n[[1]], c + cm*n[[3]]^2}}//N] \ \>", "Input", Editable->False, InitializationCell->True, AspectRatioFixed->True], Cell["\<\ RotToQuat[mat_List] := Module[{q0,q1,q2,q3,trace,s,t1,t2,t3}, trace = Sum[mat[[i,i]],{i,1,3}]; If[trace > 0, s = Sqrt[trace + 1.0]; q0 = s/2; s = 1/(2 s); q1 = (mat[[3,2]] - mat[[2,3]])*s; q2 = (mat[[1,3]] - mat[[3,1]])*s; q3 = (mat[[2,1]] - mat[[1,2]])*s, If[(mat[[1,1]] >= mat[[2,2]]) && (mat[[1,1]] >= mat[[3,3]]), (* i=0, j = 1, k = 2 *) s = Sqrt[mat[[1,1]] - mat[[2,2]] - mat[[3,3]] + 1.0]; q1 = s/2; s = 1/(2 s); q0 = (mat[[3,2]] - mat[[2,3]])*s; q2= (mat[[2,1]] + mat[[1,2]])*s; q3 = (mat[[1,3]] + mat[[3,1]])*s, If[(mat[[1,1]] < mat[[2,2]]) && (mat[[1,1]] >= mat[[3,3]]), (* i=1, j = 2, k = 0 *) s = Sqrt[mat[[2,2]] - mat[[3,3]] - mat[[1,1]] + 1.0]; q2 = s/2; s = 1/(2 s); q0 = (mat[[1,3]] - mat[[3,1]])*s; q3= (mat[[3,2]] + mat[[2,3]])*s; q1 = (mat[[2,1]] + mat[[1,2]])*s, (* Else: (mat[[1,1]] < mat[[2,2]]) && (mat[[1,1]] < mat[[3,3]]) \ *) (* i=2, j = 0, k = 1 *) s = Sqrt[mat[[3,3]] - mat[[1,1]] - mat[[2,2]] + 1.0]; q3 = s/2; s = 1/(2 s); q0 = (mat[[2,1]] - mat[[1,2]])*s; q1= (mat[[1,3]] + mat[[3,1]])*s; q2 = (mat[[3,2]] + mat[[2,3]])*s]]]; normalize[N[{q0,q1,q2,q3}]]] \ \>", "Input", Editable->False, InitializationCell->True, AspectRatioFixed->True], Cell["\<\ RotToQuatSym[mat_List] := Module[{q0,q1,q2,q3,trace,s,t1,t2,t3}, trace = Sum[mat[[i,i]],{i,1,3}]; s = Sqrt[trace + 1.0]; q0 = s/2; s = 1/(2 s); q1 = (mat[[3,2]] - mat[[2,3]])*s; q2 = (mat[[1,3]] - mat[[3,1]])*s; q3 = (mat[[2,1]] - mat[[1,2]])*s; {q0,q1,q2,q3}]\ \>", "Input", Editable->False, InitializationCell->True, AspectRatioFixed->True], Cell[TextData[StyleBox["Distances in quaternion arcs:", Evaluatable->False, AspectRatioFixed->True]], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ QLength[curveOnSphere_List] := Sum[ArcCos[curveOnSphere[[i]] . curveOnSphere[[i+1]]], {i,1,Length[curveOnSphere]-1}]\ \>", "Input", Editable->False, InitializationCell->True, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["Arrays of Quaternion frames need these: ", "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ ForceCloseQFrames[qfrms_List] := Module[{lastfrm = qfrms[[1]], thisfrm}, Table[thisfrm = qfrms[[i]]; lastfrm = If[thisfrm . lastfrm >= 0, thisfrm, - thisfrm], {i,1,Length[qfrms]}]]\ \>", "Input", Editable->False, InitializationCell->True, AspectRatioFixed->True], Cell["\<\ ForceClose2DQFrames[qrows_List] := Module[{lastrow = qrows[[1]], thisrow}, Table[thisrow = ForceCloseQFrames[qrows[[i]]]; lastrow = If[thisrow[[1]] . lastrow[[1]] >= 0, thisrow, - thisrow], {i,1,Length[qrows]}]] \ \>", "Input", Editable->False, InitializationCell->True, AspectRatioFixed->True], Cell[BoxData[ \(ForceClose3DQFrames[qplanes_List] := Module[{lastplane = qplanes[\([1]\)], thisplane}, Table[thisplane = ForceClose2DQFrames[ qplanes[\([i]\)]]; \[IndentingNewLine]lastplane = If[thisplane[\([1]\)] . lastplane[\([1]\)] \[GreaterEqual] 0, thisplane, \(-thisplane\)], {i, 1, Length[qplanes]}]]\)], 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\((1 - t)\)*p0\ + \ t*p1\)], "Input",\ Editable->False, InitializationCell->True], Cell[BoxData[ \(Slerp[p0_List, p1_List, t_]\ := \ Module[{costh\ = \ p0\ . \ p1 // Chop, \ th, \ sinth}, \[IndentingNewLine]\ \ If[costh\ > \ 0.0, costh\ = \ Chop[costh\ - 1. ]\ + 1. ]; \[IndentingNewLine]\ \ If[ costh\ < \ 0.0, costh\ = \ Chop[costh\ + 1. ]\ - 1. ]; \[IndentingNewLine]\ th\ = \ N[ArcCos[costh]]; \[IndentingNewLine]\ sinth\ = \ N[Sin[th]]; \[IndentingNewLine]If[ sinth\ == \ 0, \ \[IndentingNewLine]\ \((1 - t)\)*p0\ + \ t*p1, \[IndentingNewLine]\ \((Sin[th*\((1 - t)\)]/sinth)\)* p0\ + \((Sin[th*t]/sinth)\)*p1\ ]]\)], "Input", Editable->False, InitializationCell->True], Cell[CellGroupData[{ Cell["Spline and Spherical Splines", "Section"], Cell[BoxData[ \(CRspline[p0_, p1_, p2_, p3_, t_]\ := \n\t Lerp[Lerp[Lerp[p0, p1, t + 1], Lerp[p1, p2, t], \((t + 1)\)/ 2. ], \n\ \ \ \ \ \ \ \ \ \ \ \ \ \ Lerp[Lerp[p1, p2, t], Lerp[p2, p3, t - 1], \ \ \ \ \ \ \ \ \ \ \ t/ 2. ], \ \ \ \ \ \ \ \ \ t]\)], 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p3_, t_]\ := \n\t Slerp[Slerp[Slerp[p0, p1, t], Slerp[p1, p2, t], t], \n\ \ \ \ \ \ \ \ \ \ \ \ \ \ Slerp[Slerp[p1, p2, t], Slerp[p2, p3, t], t], t]\)], "Input", Editable->False, InitializationCell->True], Cell[BoxData[ \(SUBspline[p0_, p1_, p2_, p3_, t_]\ := \n\t Slerp[Slerp[Slerp[p0, p1, \((t + 2)\)/3. ], Slerp[p1, p2, \((t + 1)\)/3. ], \((t + 1)\)/ 2. ], \n\ \ \ \ \ \ \ \ \ \ \ \ \ \ Slerp[ Slerp[p1, p2, \((t + 1)\)/3. ], Slerp[p2, p3, t/3. ], \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ t/2. ], t]\)], "Input", Editable->False, InitializationCell->True], Cell[BoxData[ \(squad[x0_, x1_, x2_, x3_, t_]\ := \((1 - 2 t\ \((1\ - \ t)\))\)*\((\((1 - t)\)\ x0\ + \ t\ x3)\)\ + \ 2 t*\((1 - t)\) \((\((1 - t)\) x1\ + \ t\ x2)\)\)], "Input", InitializationCell->True] }, Open ]], Cell["Usage:", "Subsubsection"], Cell[CellGroupData[{ Cell[BoxData[ \(wiresphere[4, 4]\)], "Input"], Cell[BoxData[ \(wiresphere[4, 4]\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(Module[{sspl, 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