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

1. Oriented projective line, $\mathbf{T}^1$: $(x,w) \in \mathbf{R}^2_*$, with the equivalence relation $(x,w)\sim(a x,a w)\,$ for all $a>0$.
2. Oriented projective plane, $\mathbf{T}^2$: $(x,y,w) \in \mathbf{R}^3_*$, 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. $\mathbf{T}^1$ can be viewed as the union of two copies of $\mathbf{R}$, the sets (x,1) and (x,-1), plus two additional points at infinity, (1,0) and (-1,0). Likewise $\mathbf{T}^2$ can be view two copies of $\mathbf{R}^2$, (x,y,1) and (x,y,-1), plus one copy of $\mathbf{T}$ (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

x²+y²+z²=1.

Distances between two points

$p=(p_x,p_y,p_w)$

and

$q=(q_x,q_y,q_w)$

in

$\mathbf{T}^2$

can be defined as elements in $\mathbf{T}^1$

$((p_x q_w-q_x p_w)^2+(p_y q_w-q_y p_w)^2,\mathrm{sign}(p_w q_w)(p_w q_w)^2)\,.$

References

• Stolfi, Jorge (1991). Oriented Projective Geometry. Academic Press. ISBN 978-0-12-672025-9.
  From original Stanford Ph.D. dissertation, Primitives for Computational Geometry, available as .
• 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.
• Ghali, Sherif (2008). Introduction to Geometric Computing. Springer. ISBN 978-1-84800-114-5.
  Nice introduction to oriented projective geometry in chapters 14 and 15. More at authors web-site. Sherif Ghali.