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

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

These spaces can be viewed as extensions of euclidean space. ${\displaystyle \mathbf {T} ^{1}}$ 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 ${\displaystyle \mathbf {T} ^{2}}$ 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

${\displaystyle p=(p_{x},p_{y},p_{w})}$

and

${\displaystyle q=(q_{x},q_{y},q_{w})}$

in

${\displaystyle \mathbf {T} ^{2}}$

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

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

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.