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