lew@ihuxr.UUCP (Lew Mammel, Jr.) (09/28/83)
Silvio Levy made an error in his article on projective spaces.
The set of opposite point pairs on a sphere correspond to the projective
plane, not the projective 3-d space. This is obvious when you note that
each projective point is identified with { X * (a,b,c) for all X != 0 }
The euclidian coordinates of the point are given by x=a/c , y=b/c. The
points represented by (a,b,0) form the "line at infinity".
Linear transformations of the projective coordinates correspond to projective
transformations of the plane. These include rotations and translations,
as well as a set of transformations which correspond to the projection
of a horizontal plane onto a vertical plane, as in perspective drawing.
Projective geometry gets short shrift these days, but it is quite
fascinating. Some of its interesting ramifications are described in
Felix Klein's "Geometry" from his "Elementary mathematics from an advanced
viewpoint" series.
Lew Mammel, Jr. ihuxr!lew