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Re: Rotation3D, MatrixRotation3D ?

• To: mathgroup at smc.vnet.net
• Subject: [mg30392] Re: Rotation3D, MatrixRotation3D ?
• From: Jens-Peer Kuska <kuska at informatik.uni-leipzig.de>
• Date: Wed, 15 Aug 2001 01:03:59 -0400 (EDT)
• Organization: Universitaet Leipzig
• References: <9lalnl$cd3$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

ojg wrote:
> 
> Question regarding rotations.
> 
> Some of the documentation found regarding this is not as far as I can see
> complete in the documentation, at least the subject is difficult enough to
> make me unsure once not 100% clear:)
> 
> Fist, what are the defined "euler angles" in mathematica, and in what order
> are they applied?

The order of the Euler angels is fixed by it's definition.
You may look at

http://www.martinb.com/threed/scenegraph/rotations/euler.htm
> 
> Second, of which side of the vector is the rotational matrix multiplied ?

As usual matrix.vector 

> 
> Third, is there a mathematica way to rotate around an abitrary rotational
> axis? If not, what would the mathematica matix be for this?

rotationmatrix[axis_?VectorQ,theta_?NumericQ] :=
  Module[{nx,ny,nz,st,ct,norm},
    norm=Sqrt[Dot[axis,axis]];
    If[norm<$MachineEpsilon, Return[IndentityMatrix[3]] ];
    {nx,ny,nz} = N[axis/norm];
   {ct,st}=N[{Cos[theta],Sin[theta]}];
	   N[{{ nx^2 + (1 - nx^2)*ct,nx*ny*(1 - ct) + nz*st,nx*nz*(1 - ct) -
ny*st},
 	     { nx*ny*(1 - ct) - nz*st,ny^2 + (1 - ny^2)*ct,ny*nz*(1 - ct) +
nx*st},
	     { nx*nz*(1 - ct) + ny*st,ny*nz*(1 - ct) - nx*st,nz^2 + (1 -
nz^2)*ct}}]
  ]

MathGL3d has the MVRotate3D[] function to do vector/axis rotations.


> 
> My problem to solve is as follows: Given three rotational angles (a,b,c)
> applied in order to the following three rotational axes: Y axis, X axis, Z
> axis. (usual right hand system). This rotation applied to any vector v will
> give you a vector V (first Y rotation applied on v, etc).

And for what do you need Euler angles ?

> 
> Now, given a rotational matrix with pure numerical values in, I need to find
> the three angles, and I need a general formulae for this solution taking
> care of the special cases.

This can't exist. Because for an arbitary matrix it will not possible to
find a
solution. The matrix must be orthogonal R^T R=1. 

Regards
  Jens