On Tuesday 08 March 2011 06:00:56 Fred Lunnon wrote (to just me rather than math-fun, but that was an error): [me:]
A(Cx+d)+b = (AC)x + (Ad+b). In what sense are these transformations not closed under composition?
[Fred, whole message kept for the sake of context:]
This (on the face of it) perfectly reasonable objection made me realise how accustomed I have become to thinking about these matters in terms of the symmetries acting on the space, to the extent that it did not occur to me that there might be more than one way to define "composition" in this context.
It also points up very nicely one of the major problems concealed in the matrix times vector representation of geometric transformations: that it breaks down as soon as you want to transform anything else besides a point, such as another isometry.
In 2-space say, the translation by T of a rotation R around the origin equals rotation about the translated centre, represented by T^{-1} R T --- which now cannot expressed in the form A x + b for A in GL(2).
1. Rotation about a nonzero point *is* of the form x -> Ax+b. 2. The definition of an affine transformation does not require that A be in GL(n). What am I missing here?
The situation deteriorates further in 3-space and above, where even the axial coline is not canonically representable.
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*?
The interpretation that symmetries act on other symmetries becomes inevitable when the geometry is defined in a Kleinian fashion: for instance, it seems clear to me that the author of the discussion at http://en.wikipedia.org/wiki/Affine_geometry is also (implicitly) taking this view. I wonder too if some of the difficulty I have experienced with the use of the term "affine" stems from the same dichotomy.
Incidentally, for anyone (like most practitioners of computer graphics) acustomed to thinking of subspaces (particularly points) as fundamental, but transformations as a derivative concept, to reverse the bias demands a substantial mental realignment. I can still recall the shocking discovery that planes, lines, points could all be represented perfectly well as if they were instead the reflections, half-turns etc. leaving them pointwise fixed --- a fundamental feature of the geometric algebra approach.
-- g