Before we introduce projective geometry let us briefly consider
spherical geometry (
), which is the geometry on the surface of a sphere.
: If one begins at a point in
travels straight ahead on the surface, one will trace out a
great circle. Viewed as a set in
this is the intersection
with a plane through the origin. We will call
this great circle a line in
be a unit vector. Then,
is the line with pole
are antipodal if
cannot be parallel. Any two lines intersect at a
pair of antipodal points.