I did myself no favours earlier by writing "composition" when I meant "application". Thinking about how this blunder arose uncovers yet another wrinkle of traditional point-based coordinates, with repercussions down the line: matrix-times-matrix product represents _composition_ of symmetries; yet matrix-times-vector represents _application_ of a symmetry to a point! On 3/8/11, Gareth McCaughan <gareth.mccaughan@pobox.com> wrote:
1. Rotation about a nonzero point *is* of the form x -> Ax+b.
Aha! That's quite right --- and the analogue is true for n-space --- any Euclidean isometry can be expressed as the composition of at most n+1 copoint (hyperplane) reflections, each further decomposable as T^{-1} F T for translation T and reflection F fixing the origin.
2. The definition of an affine transformation does not require that A be in GL(n).
What am I missing here?
It's true that the wikipedia article is not explicit concerning dimension. But if you're implying A could in GL(n+1) initially rather than GL(n), what would be the point of later adding b for translations? You would already have the entire projective group, which is (presumably) too large! The article does go on to explain how enlarging the coordinate vector x to a homogeneous projective (n+1)-vector permits translation to be incorporated into a GL(n+1) matrix.
What do you mean by "not canonically representable", and why is it a problem that it isn't, and why is that a problem *with the notion of affine transformation*?
It's a problem with point-based representation, which offers no unique way to represent colines such that it is possible immediately to decide whether two given colines are equal. For example, the best you can do for lines in 3-space is to store them as pairs of points, then tinker around with the rank of all 4 vectors. In GL(n+1) there is the GA-style option of representing any subspace by the isometry (reflection, half-turn rotation, etc) of which it forms the axis --- which now transforms via conjugation rather than multiplication. For the sake of uniformity, one would then prefer also to represent points in this fashion, and drop matrix-times-vector application altogether! However even so, there are apparently no simple algorithms for extracting natural metrical invariants, such as the angle between two lines.
g
The "affine" notion does perhaps begin to look less arbitrary; though I'm still in the dark about what use it might be ... What would a canonical example of a projective transformation which is not affine --- a perspectivity, perhaps? Fred Lunnon