Oriented projective geometry

Oriented projective geometry is an oriented version of real 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 R ∗ n {displaystyle mathbf {R} _{*}^{n}} be the set of elements of R n {displaystyle mathbf {R} ^{n}} excluding the origin. These spaces can be viewed as extensions of euclidean space. T 1 {displaystyle mathbf {T} ^{1}} can be viewed as the union of two copies of R {displaystyle mathbf {R} } , the sets (x,1) and (x,-1), plus two additional points at infinity, (1,0) and (-1,0). Likewise T 2 {displaystyle mathbf {T} ^{2}} can be view two copies of R 2 {displaystyle mathbf {R} ^{2}} , (x,y,1) and (x,y,-1), plus one copy of T {displaystyle mathbf {T} } (x,y,0). An alternative way to view the spaces is as points on the circle or sphere, given by the points (x,y,w) with