Re: Multidimensional rotations

On Jul 24, 10:25 pm, ekpuz.b...@xxxxxxxxxxx wrote:
snip
If the answer is not forthcoming from someone, I'd appreciate
speculation or suggestions as to what sorts of things might be good to
try, or how many rotations might be necessary (I'm guessing N-M).

I'd also be open to hearing if anyone has any ideas for a
straightforward way to generate a set of M basis vectors in N
dimensional space (N>M) where the covariance between the different
vectors is minimized.  While I think this will be a multidemensional
analog to a tetrahedron, if it is something else I'm open to
suggestions or pointers.

Have a look at

http://en.wikipedia.org/wiki/Geometric_algebra

Geometric (Clifford) algebras cl[p,q] are built from a set of
"vectors" e[i] (I call them "monovectors" because "vector" is
ambiguous) that are orthogonal e[i]\dot\.e[j]=0, antisymmetric e[i]
\wedge\e[j] = - e[j]\wedge\e[i], with p that square to +1 and q that
square to -1. They define a space with p+q directions; the products of
2, 3, ... monovectors are called bivectors, trivectors, etc.. These
can be thought of as areas, volumes, ... and they generalises
geometry, algebra, analysis etc. to any number of dimensions. General
rotations are pre- and post-multiplication by half angles. M
corresponds to a subset of an N-directional algebra. Each monovector
defines a scale for the appropriate direction, and N-M are "zeroed-
out" in the subset. Does orthogonality meet your "minimum covariance"
concept?
Be warned. Geometric algebra has idiosyncratic and confusing
nomenclature. Multivector means a list of elements (a vector in
Mathematica) It can be graded into scalar, monovectors, bivectors, ...
pseudovector. Each grade is called a blade. Then there are rotors,
spinors, versors ... .

RogerB
.

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