Re: Solve & RotationMatrix

• To: mathgroup at smc.vnet.net
• Subject: [mg76165] Re: Solve & RotationMatrix
• From: Mathieu G <ellocomateo at free.fr>
• Date: Thu, 17 May 2007 05:56:07 -0400 (EDT)
• References: <f21g0q\$7d6\$1@smc.vnet.net> <f23qq6\$oa3\$1@smc.vnet.net> <31173113.1179139106764.JavaMail.root@m35> <f2ei6m\$spd\$1@smc.vnet.net>

```DrMajorBob a =E9crit :
> 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> wro=
te:
>
>> 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 Vect=
N.
>> Regards,
>> Mathieu
>>
>>
>
>
>
Thank you I will be working with that in the next few hours!

```

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