Re: Eulerian angles
- To: mathgroup at smc.vnet.net
- Subject: [mg42674] Re: [mg42668] Eulerian angles
- From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
- Date: Sun, 20 Jul 2003 06:20:53 -0400 (EDT)
- Sender: owner-wri-mathgroup at wolfram.com
On Saturday, July 19, 2003, at 09:20 AM, Selwyn Hollis wrote: > Some 5 or 6 years ago, I asked a question in MathGroup about the "Euler > angles" that are used by RotateShape. Apparently physicists know all > about this stuff, but I still have almost no feeling for what these > angles are about. So I thought I'd issue this challenge: > > Create *the* graphic illustrating the Euler angles that ought to be in > the Mathematica Book --- hopefully understandable by a hack > mathematician and his calculus students. > > The winner will receive glowing praise and thanks in a soon-to-be > published book. > > ----- > Selwyn Hollis > http://www.math.armstrong.edu/faculty/hollis > > > For a mathematician the easiest way to understand what Euler angles are about is by considering compact Lie groups and their Lie algebras (provided you know about these things). The essential fact are: 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 also called the Eulerian angles in SU(2). Finally the natural homomorphism SU(2)->SO(3) is onto (a double covering in fact) so we get the "usual" Eulerian angles in SO(3) from those in SO(2). All this is described in greater detail in the excellent book on Linear algebra by Kostrikin and Manin. As for graphical illustration of the meaning of Eulerian angles, David Park has an excellent package and notebook that does this and I am sure he will gladly send it to you (and will the glowing praise and mention in your book). Andrzej Kozlowski Yokohama, Japan http://www.mimuw.edu.pl/~akoz/ http://platon.c.u-tokyo.ac.jp/andrzej/