Re: Re: Solve & RotationMatrix

*To*: mathgroup at smc.vnet.net*Subject*: [mg76071] Re: [mg76002] Re: Solve & RotationMatrix*From*: DrMajorBob <drmajorbob at bigfoot.com>*Date*: Wed, 16 May 2007 05:16:36 -0400 (EDT)*References*: <f21g0q$7d6$1@smc.vnet.net> <f23qq6$oa3$1@smc.vnet.net> <31173113.1179139106764.JavaMail.root@m35>*Reply-to*: drmajorbob at bigfoot.com

Here's a much simpler (and probably more robust) solver, using the fact that the eigenvalues of a rotation matrix are {1, Cos[theta] + I Sin[theta], Cos[theta] - I Sin[theta]} Add those, subtract 1, divide by 2, and take ArcCos to get theta. As before, the real eigenvector is the axis of rotation. (* code *) Clear[realQ, realSelect, randomRotationMatrix, angleVector] realQ[z_?NumericQ] := Re[z] == z realQ[v_?VectorQ] := VectorQ[v, realQ] realQ[v_?MatrixQ] := MatrixQ[v, realQ] realSelect[m_?MatrixQ] := First@Select[m, realQ, 1] randomRotationMatrix[] := Module[{t, v}, t = RandomReal[{-Pi, Pi}]; v = Normalize@Table[RandomReal[], {i, 3}]; {"angle" -> t, "axis" -> v, "rotation matrix" -> RotationMatrix[t, v]}] angleVector[target_?MatrixQ] /; realQ[target] := Module[{values, vectors}, {values, vectors} = Eigensystem[target]; {"angle" -> ArcCos[(Plus @@ values - 1)/2], "axis" -> realSelect@vectors} ] (*create a test problem*) randomRotationMatrix[]//TableForm target="rotation matrix"/.%; angle->2.25948 axis->{0.152625,0.442375,0.883747} rotation = matrix->{{-0.597426,-0.571897,0.562152},{0.792751,-0.315459,0.521567},{-0.120946,0.757244,0.641836}} (*solve for the angle and vector*) angleVector[target] {angle->2.25948,axis->{0.152625,0.442375,0.883747}} Bobby On Mon, 14 May 2007 04:49:33 -0500, Mathieu G <ellocomateo at free.fr> wrote: > CKWong.P at gmail.com a =E9crit : >> Obviously, you need to provide the function RotationMatrix[ angle, >> axisVector ] for the algorithm to work. >> >> On the other hand, matrices for rotations about the Cartesian axes, >> i.e., RotX,RotY, & RotZ, can be written done directly. Why not do so? >> >> > Hi, > Thank you for your reply. > I am not interested in a simple rotation matrix around an axis, but in > finding the angles corresponding to a 3D rotation matrix. > I compute VectN which is the vector normal to my surface, and I am > interested in getting the rotation parameters that bring VectY to VectN. > Regards, > Mathieu > > -- DrMajorBob at bigfoot.com