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 $\mathbb_^n$ be the set of elements of $\mathbb^n$ excluding the origin.
#Oriented projective line, $\mathbb^1$: $(x,w) \in \mathbb^2_*$, with the equivalence relation $(x,w)\sim(a x,a w)\,$ for all $a>0$.
#Oriented projective plane, $\mathbb^2$: $(x,y,w) \in \mathbb^3_*$, 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. $\mathbb^1$ can be viewed as the union of two copies of $\mathbb$, the sets (''x'',1) and (''x'',-1), plus two additional points at infinity, (1,0) and (-1,0). Likewise $\mathbb^2$ can be viewed as two copies of $\mathbb^2$, (''x'',''y'',1) and (''x'',''y'',-1), plus one copy of $\mathbb$ (''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

:''x''²+''y''²+''w''²=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
:$\mathbb T^n=\=\.$
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 $\mathbb^2$ can be defined as elements
:$((p_x q_w-q_x p_w)^2+(p_y q_w-q_y p_w)^2,\mathrm(p_w q_w)(p_w q_w)^2)$
in $\mathbb^1$.

Oriented complex projective geometry

Let ''n'' be a nonnegative integer. The oriented complex projective space $^n_$ is defined as
:$^n_=\=\$. Here, we write $S^1$ to stand for the 1-sphere.

See also

* Variational analysis

Notes

References

*
From original Stanford Ph.D. dissertation, ''Primitives for Computational Geometry'', available a


*
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.
*{{cite book , last1=Werner , first1=Tomas , title=Proceedings Ninth IEEE International Conference on Computer Vision , chapter=Combinatorial constraints on multiple projections of set points , date=2003 , pages=1011–1016 , doi=10.1109/ICCV.2003.1238459 , isbn=0-7695-1950-4 , s2cid=6816538 , chapter-url=https://ieeexplore.ieee.org/document/1238459 , access-date=26 November 2022

Projective geometry