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
- Oriented projective line,
T 1 ( x , w ) ∈ R ∗ 2 ( x , w ) ∼ ( a x , a w ) for all a > 0 . - Oriented projective plane,
T 2 ( x , y , w ) ∈ R ∗ 3 ( x , y , w ) ∼ ( a x , a y , a w ) for all a > 0 .
These spaces can be viewed as extensions of euclidean space.
An alternative way to view the spaces is as points on the circle or sphere, given by the points (x,y,w) with
x2+y2+z2=1.Distances between two points
and
in
can be defined as elements in