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 $$\mathbb{R}_{*}^n$$ be the set of elements of $$\mathbb{R}^n$$ excluding the origin.
 * 1) Oriented projective line, $$\mathbb{T}^1$$: $$(x,w) \in \mathbb{R}^2_*$$, with the equivalence relation $$(x,w)\sim(a x,a w)\,$$ for all $$a>0$$.
 * 2) Oriented projective plane, $$\mathbb{T}^2$$: $$(x,y,w) \in \mathbb{R}^3_*$$, with $$(x,y,w)\sim(a x,a y,a w)\,$$ for all $$a>0$$.

These spaces can be viewed as extensions of euclidean space. $$\mathbb{T}^1$$ can be viewed as the union of two copies of $$\mathbb{R}$$, the sets (x,1) and (x,-1), plus two additional points at infinity, (1,0) and (-1,0). Likewise $$\mathbb{T}^2$$ can be viewed as two copies of $$\mathbb{R}^2$$, (x,y,1) and (x,y,-1), plus one copy of $$\mathbb{T}$$ (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


 * x2+y2+w2=1.

Oriented real projective space
Let n be a nonnegative integer. The (analytical model of, or canonical) oriented (real) projective space or (canonical) two-sided projective space $$\mathbb T^n$$ is defined as
 * $$\mathbb T^n=\{\{\lambda Z:\lambda\in\mathbb R_{>0}\}:Z\in\mathbb R^{n+1}\setminus\{0\}\}=\{\mathbb R_{>0}Z:Z\in\mathbb R^{n+1}\setminus\{0\}\}.$$

Here, we use $$\mathbb T$$ to stand for two-sided.

Distance in oriented real projective space
Distances between two points $$p=(p_x,p_y,p_w)$$ and $$q=(q_x,q_y,q_w)$$ in $$\mathbb{T}^2$$ can be defined as elements
 * $$((p_x q_w-q_x p_w)^2+(p_y q_w-q_y p_w)^2,\mathrm{sign}(p_w q_w)(p_w q_w)^2)$$

in $$\mathbb{T}^1$$.

Oriented complex projective geometry
Let n be a nonnegative integer. The oriented complex projective space $${\mathbb{CP}}^n_{S^1}$$ is defined as
 * $${\mathbb{CP}}^n_{S^1}=\{\{\lambda Z:\lambda\in\mathbb R_{>0}\}:Z\in\mathbb C^{n+1}\setminus\{0\}\}=\{\mathbb R_{>0}Z:Z\in\mathbb C^{n+1}\setminus\{0\}\}$$. Here, we write $$S^1$$ to stand for the 1-sphere.