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 ${\displaystyle \mathbb {R} _{*}^{n}}$ be the set of elements of ${\displaystyle \mathbb {R} ^{n}}$ excluding the origin.

1. Oriented projective line, ${\displaystyle \mathbb {T} ^{1}}$: ${\displaystyle (x,w)\in \mathbb {R} _{*}^{2}}$, with the equivalence relation ${\displaystyle (x,w)\sim (ax,aw)\,}$ for all ${\displaystyle a>0}$.
2. Oriented projective plane, ${\displaystyle \mathbb {T} ^{2}}$: ${\displaystyle (x,y,w)\in \mathbb {R} _{*}^{3}}$, with ${\displaystyle (x,y,w)\sim (ax,ay,aw)\,}$ for all ${\displaystyle a>0}$.

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

Let n be a nonnegative integer. The (analytical model of, or canonical¹) oriented (real) projective space or (canonical²) two-sided projective³ space ${\displaystyle \mathbb {T} ^{n}}$ is defined as

${\displaystyle \mathbb {T} ^{n}=\{\{\lambda Z:\lambda \in \mathbb {R} _{>0}\}:Z\in \mathbb {R} ^{n+1}\setminus \{0\}\}=\{\mathbb {R} _{>0}Z:Z\in \mathbb {R} ^{n+1}\setminus \{0\}\}.}$⁴

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

The Euclidean distance between two points ${\displaystyle p=(p_{x},p_{y},p_{w})}$ and ${\displaystyle q=(q_{x},q_{y},q_{w})}$ in ${\displaystyle \mathbb {T} ^{2}}$ can be defined as elements

${\displaystyle ({\sqrt {(p_{x}q_{w}-q_{x}p_{w})^{2}+(p_{y}q_{w}-q_{y}p_{w})^{2}}},p_{w}q_{w}))}$

in ${\displaystyle \mathbb {T} ^{1}}$.⁵

The squared Euclidean distance is then:

${\displaystyle ((p_{x}q_{w}-q_{x}p_{w})^{2}+(p_{y}q_{w}-q_{y}p_{w})^{2},\mathrm {sign} (p_{w}q_{w})(p_{w}q_{w})^{2})}$

Let n be a nonnegative integer. The oriented complex projective space ${\displaystyle {\mathbb {CP} }_{S^{1}}^{n}}$ is defined as

${\displaystyle {\mathbb {CP} }_{S^{1}}^{n}=\{\{\lambda Z:\lambda \in \mathbb {R} _{>0}\}:Z\in \mathbb {C} ^{n+1}\setminus \{0\}\}=\{\mathbb {R} _{>0}Z:Z\in \mathbb {C} ^{n+1}\setminus \{0\}\}}$.⁶ Here, we write ${\displaystyle S^{1}}$ to stand for the 1-sphere.