Re: [math-fun] Reflections on Orientation
Fred wrote: << . . . It's now evident that a major source of confusion was that [coming from a computational geometry direction --- orientation?] I'm concerned rather with orientation of a subspace within the whole space. If we consider more general geometries --- Moebius / conformal / Poincar\'e, Lie sphere / physicist's conformal, and so on --- the Lie group corresponding to Euclidean isometries has 4 connected components rather than 2. On the other hand, orientation of subspaces does not appear to be any more difficult than before. So the idea occurs to me that perhaps the solution to my problem is to consider instead a 4-valued "sub-orientation"? Why should this be justified?
If a concrete math problem or situation were stated, then I might understand what is being discussed here. Like: * What is it about the "orientation of a subspace within the whole space" that is of interest to you? What kinds of spaces? * With the "4 connected components" thing -- what is one concrete example of what you mean here? * What is it about "orientation of subspaces" that may or may not be "difficult" ? --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele
On 1/27/08, Dan Asimov <dasimov@earthlink.net> wrote:
If a concrete math problem or situation were stated, then I might understand what is being discussed here.
Like:
* What is it about the "orientation of a subspace within the whole space" that is of interest to you? What kinds of spaces?
Traditional 3-space projective geometry together with its representation via homogeneous point coordinates, as applied to computer-aided Graphics, Vision, Robotics, Design etc, is deficient in two notable respects: it fails to treat k-flat subspaces on an equal footing with 0-flat points; and it fails to distinguish the behaviour of various orientable properties of those flats under the action (particularly) of isometries. Examples of such properties are: a plane segment may be decorated with lettering, and we need to know whether or not this will be illegibly reversed; a plane segment may be painted with distinct colours on the two sides, and we need to know which colour will be visible; a line segment may be decorated with an arrowhead, and we need to know at which end this will lie; a line segment may be the axis of a rotation, and we need to know in which sense a positive angle will turn. This topic has been considered before, for instance by [Sto91] Jorge~Stolfi "Oriented Projective Geometry" Academic Press (1991) and appeared to have successfully been formalised in a fashion which associates an extra binary value with a flat, extending the binary notion that already appears familiar in other contexts. [I don't have this by me at the moment, so cannot check details, and must rely on higly unreliable memory.] Indeed, this approach appears to work perfectly satisfactorily provided only proper isometries are involved [or incidentally if k = n]: the traditional representation of a flats by quotients of (Pluecker) coordinate vectors over nonnegative reals has to be replaced by quotients over positive reals; then a change of sign indicates a change of orientation.
* With the "4 connected components" thing -- what is one concrete example of what you mean here?
Misprint --- see previous posting --- standard Lie group stuff.
* What is it about "orientation of subspaces" that may or may not be "difficult" ?
Unfortunately once improper isometries are admitted --- as the example of the transparent sheet decorated with "Happy New Year" in letters with green fronts and red backs illustrates --- the binary solution breaks down for planes in 3-space. A similar example involving a spinning top with a handle at one end of its axle illustrates the same difficulty for lines in 3-space. In both cases, when suitably reflected in a mirror, these occupy the same locus but only _one_ out of the _two_ orientable properties has altered. It appears finally that in order to capture orientation of Euclidean flats under the action of improper isometries, a base-4 variable is required: one bit indicating whether an n-space isometry (fixing the flat locus) acts properly on the flat (lettering, arrowhead); and another bit indicating whether it acts properly on the perpendicular flat (colour, sense). The resulting inconvenience may in practice be minor, and [like many another such] overcome when it arises by a suitable kludge. Nonetheless, it seems to me extraordinary that such an obvious matter appears not only unrecognised, but [perhaps for reasons unconnected with geometry] in some way unrecognisable. Fred Lunnon
I bought a laptop during boxing week sale. You should get one so you can read my CD's.
On 1/28/08, Fred lunnon <fred.lunnon@gmail.com> wrote:
.... Indeed, this approach appears to work perfectly satisfactorily provided only proper isometries are involved [or incidentally if k = n]: the traditional representation of a flats by quotients of (Pluecker) coordinate vectors over nonnegative reals has to be replaced by quotients over positive reals; then a change of sign indicates a change of orientation. ...
Duh! For "nonnegative" read "nonzero". WFL
Here's a practical instance of inconsistant orientation, which caused me some head-scratching at the time I first encountered it --- the algorithm employed for {\sl back face culling} of those faces of a convex polyhedron invisible at a given viewpoint when projected from {3-space} onto a given viewplane {2-space}. During the construction of the polyhedron from its vertex points, the polygons defining the faces all must be traversed in the same sense (say anticlockwise) when viewed from the exterior, for the purposes of computing face planes from vertices. The distances of faces visible at the viewpoint will then all have the same sign (say positive), and of faces invisible the opposite sign (say negative). However, visibility is a matter of P-orientation --- whether the external side of the plane is towards the viewer; but cyclic order of vertices is a matter of R-orientation --- whether an internal triangle is ordered anticlockwise. As a result, if the polyhedron becomes reflected in a plane mirror, it will also appear to the viewer to have mysteriously been turned inside out, unless the algorithm has been patched up by negating the distance signs. Fred Lunnon
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