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). > > > > > > > > > > > > "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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