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 $(x,w)\sim (ax,aw)$ for all $a>0$.
2. Oriented projective plane, ${\mathbb{𝕋}}^2$: $(x,y,w)\in {\mathbb{ℝ}}_{*}^3$, with $(x,y,w)\sim (ax,ay,aw)$ 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 canonicalTemplate:Sfn) oriented (real) projective space or (canonicalTemplate:Sfn) two-sided projectiveTemplate:Sfn space ${\mathbb{𝕋}}^n$ is defined as

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

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

$((p_x{q}_w-q_x{p}_w{)}^2+(p_y{q}_w-q_y{p}_w{)}^2,\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}(p_w{q}_w)(p_w{q}_w{)}^2)$

in ${\mathbb{𝕋}}^1$.Template:Sfn

Oriented complex projective geometry

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

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

See also

• Variational analysis

References

• 
  From original Stanford Ph.D. dissertation, Primitives for Computational Geometry, available as [1].
• 
  Nice introduction to oriented projective geometry in chapters 14 and 15. More at author's website. Sherif Ghali.
• A. G. Oliveira, P. J. de Rezende, F. P. SelmiDei An Extension of CGAL to the Oriented Projective Plane T2 and its Dynamic Visualization System, 21st Annual ACM Symp. on Computational Geometry, Pisa, Italy, 2005.