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