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Re: Re: Rotation3D, MatrixRotation3D ?
*To*: mathgroup at smc.vnet.net
*Subject*: [mg30454] Re: Re: Rotation3D, MatrixRotation3D ?
*From*: "ojg" <ole.jonny.gjoen at hitecvision.com>
*Date*: Wed, 22 Aug 2001 01:41:50 -0400 (EDT)
*References*: <9ligov$iru$1@smc.vnet.net>
*Sender*: owner-wri-mathgroup at wolfram.com
Thanks a lot for your input. Most helpful!
But Please restate euler theorem, I dont seem to get the boundaries
right:...
Johnny
...:
The main point about Euler angles is the following theorem due to, not
surprisingly, Euler:
Theorem (Euler):
Any special orthogonal matrix B can be expressed in the form:
B = MatrixForm[\[CapitalPhi]].MatrixForm[\[CapitalTheta]].MatrixForm[\
\[CapitalPsi]]
where 0<\[Phi]<0,0<\[Theta]<0,-Pi<\[Psi]<0. If \[Theta] is not 0 than
the expression is unique.
(\[Phi],\[Theta],\[Psi] are the Euler angles).
"Andrzej Kozlowski" <andrzej at tuins.ac.jp> wrote in message
news:9ligov$iru$1 at smc.vnet.net...
>
> On Wednesday, August 15, 2001, at 02:04 PM, Gianluca Cruciani wrote:
>
> > "ojg" <ole.jonny.gjoen at hitecvision.com> wrote in message
> > news:<9lalnl$cd3$1 at smc.vnet.net>...
> >> 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?
> >>
> >
> > There are a number of conventions about Euler angles, I know at least
> > two of them. You can read the one used by Mathematica4 in the Help
> > Browser, searching for the "Geometry`Rotations`" package.
>
> ???
>
> Euler angles are based on a simple mathematical idea of which, for some
> reason, physicists make a great deal more than it is worth.
>
> To use Euler angles and rotation matrices we load the package
> "Geometry`Rotations`" (which of course Johnny must have known about
> otherwise he would not have asked his question. And he is quite right
> that the explanation of this matter in the Help Browser is virtually
> non-existent).
>
> Needs["Geometry`Rotations`"]
>
> Let's define three matrices.
>
> \[CapitalPhi] = RotationMatrix3D[-\[Phi], 0, 0]; \[CapitalTheta] =
> RotationMatrix3D[0, -\[Theta], 0]; \[CapitalPsi] =
> RotationMatrix3D[-\[Psi], 0, 0];
>
> The main point about Euler angles is the following theorem due to, not
> surprisingly, Euler:
>
> Theorem (Euler):
>
> Any special orthogonal matrix B can be expressed in the form:
>
> B = MatrixForm[\[CapitalPhi]].MatrixForm[\[CapitalTheta]].MatrixForm[\
> \[CapitalPsi]]
>
> where 0<\[Phi]<0,0<\[Theta]<0,-Pi<\[Psi]<0. If \[Theta] is not 0 than
> the expression is unique.
>
> (\[Phi],\[Theta],\[Psi] are the Euler angles).
>
> This is simply a statement in linear algebra and can be proved without
> any mention of rotations (see below). It can be interpreted as a
> statement about rotations in basically two different ways (in addition
> there are various conventions concerning which the letters denoting the
> angles , their order , signs of angles etc). One way is by using a fixed
> coordinate system. In this interpretation the statement asserts that any
> rotation can be expressed as a product of three rotations, about (say)
> the z axis, x axis and again z-axis. The rotations are composed in the
> same order as the matrices. The other interpretation, which seems to be
> popular with physicists, uses variable coordinates, or what they call
> "body coordinates". If you use this interpretation the order in which
> rotations are applied is the reverse of the order in which the matrices
> are multiplied.
>
> Of course the rotations need not involve coordinate axes at all. In
> general any rotation can be expressed as a product of three rotations
> about just two (not three!) perpendicular axes passing through the
> center of the rotation.
>
> The real reason why all this works can be found in the combination of
> the following facts. First, the Lie algebra su(2) (skew hermitian
> matrices with trace 0) of the Lie group SU(2) (special unitary matrices)
> is a three dimensional vector space and thus has a basis consisting of
> three vectors, say v1, v2, v3. The exponential map su(2)->SU(2) is
> surjective, so every element of SU(2) can be written as a product
> MatrixExp[\[Phi] v1]*MatrixExp[\[Theta] v2]**MatrixExp[\[Psi]*v3]. The
> real numbers \[Phi],\[Theta],\[Psi] are sometimes called the Euler
> angles in SU(2).
> Finally the natural homomorphism SU(2)->SO(3) is onto (a double covering
> in fact) so we get the usual Euler angles in SO(3) from those in SO(2).
>
>
> Andrzej Kozlowski
> Toyama International University
> JAPAN
> http://platon.c.u-tokyo.ac.jp/andrzej/
>
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