Re: [math-fun] A rhinoceros' pancreas
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
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
"Fl" == Fred lunnon <fred.lunnon@gmail.com> writes:
Fl> What would a canonical example of a projective transformation which Fl> is not affine --- a perspectivity, perhaps? If I read that correctly, you ask, in effect, for an example of what can be done with rational B-Splines but not by non-rational (to put the q in a form understandable by (computer) graphics artists. Yes? The classic answer is to transform a model of some object in a manner consistant with what it would look like in real life. A railroad vanishing to the horizon is a typical choice. The opening text of Star Wars also is a common example. In short, in the graphics world, affine refers to transforms which can be done with non-rational polynomial parametric models in R²P or R³P. I don't know whether that matches the orignal meaning of affine, but that *is* what it has come to mean in the graphics world. -JimC -- James Cloos <cloos@jhcloos.com> OpenPGP: 1024D/ED7DAEA6
From the matrix definition in wikipedia, the freedom of the n-space affine group [its dimension as a manifold in projective matrix space] equals n^2 + n = ( (n+1)^2 - 1 ) - n , suggesting that geometrically, the affine group is simply the projective subgroup fixing the co-point (hyperplane) at infinity. So why would it not be defined this way in the first place?
However the first interpretation seems inconsistent with Cederberg, who defines her "affine" group to fix an "absolute" quadric (whatever that precisely means). The constraint would reduce the freedom to (at most) ( (n+1)^2 - 1 ) - ( (n+2)(n+1)/2 - 1 ) = (n+1)n/2 , consistent with the Euclidean group --- as I originally assumed! Then a whole new can of worms is opened below --- a third proposal apparently inconsistent with either of the above. I rest my case --- affine faffine! WFL On 3/9/11, James Cloos <cloos@jhcloos.com> wrote:
... If I read that correctly, you ask, in effect, for an example of what can be done with rational B-Splines but not by non-rational (to put the q in a form understandable by (computer) graphics artists. Yes?
The classic answer is to transform a model of some object in a manner consistant with what it would look like in real life. A railroad vanishing to the horizon is a typical choice.
The opening text of Star Wars also is a common example.
In short, in the graphics world, affine refers to transforms which can be done with non-rational polynomial parametric models in R²P or R³P.
I don't know whether that matches the orignal meaning of affine, but that *is* what it has come to mean in the graphics world.
-JimC
participants (3)
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Fred lunnon -
Gareth McCaughan -
James Cloos