Re: Re: Re: Rotation3D, MatrixRotation3D ?
- To: mathgroup at smc.vnet.net
- Subject: [mg30465] Re: [mg30454] Re: Re: Rotation3D, MatrixRotation3D ?
- From: Andrzej Kozlowski <andrzej at tuins.ac.jp>
- Date: Thu, 23 Aug 2001 02:15:30 -0400 (EDT)
- Sender: owner-wri-mathgroup at wolfram.com
Sorry about that! I don't know how I managed to type such nonsense
(there are rather few angles angles between 0 and 0 :) ).
Here is a re-statement of the theorem.
\[CapitalPhi] = RotationMatrix3D[-\[Phi], 0, 0]; \[CapitalTheta] =
RotationMatrix3D[0, -\[Theta], 0]; \[CapitalPsi] =
RotationMatrix3D[-\[Psi], 0, 0];
Any special orthogonal matrix B can be expressed in the form:
B = MatrixForm[\[CapitalPhi]].MatrixForm[\[CapitalTheta]].MatrixForm[\
\[CapitalPsi]]
where -Pi<\[Phi]<Pi,0<\[Theta]<Pi,-Pi<\[Psi]<Pi. If \[Theta] is not 0 or
Pi than
the expression is unique.
(in the special case \[Theta]=0 or Pi one of the angles \[Phi] or \[Psi]
can be arbitrary)
Note that the matrix A= B/.MatrixForm[x_]->x is the same as
RotationMatrix3D[-\[Psi], -\[Theta], -\[Phi]].
On Wednesday, August 22, 2001, at 02:41 PM, ojg wrote:
> 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).
>
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> "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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