Oriented projective geometry


Oriented projective geometry is an oriented version of real projective geometry.
Whereas the real projective plane describes the set of all unoriented lines through the origin in R3, the oriented projective plane describes lines with a given orientation. There are applications in computer graphics and computer vision where it is necessary to distinguish between rays light being emitted or absorbed by a point.
Elements in an oriented projective space are defined using signed homogeneous coordinates. Let be the set of elements of excluding the origin.
  1. Oriented projective line, :, with the equivalence relation for all.
  2. Oriented projective plane, :, with for all.
These spaces can be viewed as extensions of euclidean space. can be viewed as the union of two copies of, the sets and, plus two additional points at infinity, and. Likewise can be viewed as two copies of, and, plus one copy of .
An alternative way to view the spaces is as points on the circle or sphere, given by the points with

Oriented real projective space

Let n be a nonnegative integer. The oriented projective space or two-sided projective space is defined as
Here, we use to stand for two-sided.

Distance in oriented real projective space

Distances between two points and in can be defined as elements
in.

Oriented complex projective geometry

Let n be a nonnegative integer. The oriented complex projective space is defined as