       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>

```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,
> 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

```

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