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 canonical) oriented (real) projective space or (canonical) two-sided projective space $\mathbb {T} ^{n}$ is defined as

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

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

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

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

Oriented complex projective geometry

See also: Complex projective space

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

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

See also

• Variational analysis

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 [1].
• 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 author's website. Sherif Ghali.
• Yamaguchi, Fujio (2002). Computer-aided Geometric Design: A Totally Four-dimensional Approach. Springer. ISBN 978-4-431-68007-9.
• Below, Alexander; Krummeck, Vanessa; Richter-Gebert, Jurgen (2003). "Complex matroids: phirotopes and their realizations in rank 2". In Aronov, Boris; Basu, Saugata; Pach, Janos; Sharir, Micha (eds.). Discrete and Computational Geometry: The Goodman–Pollack Festschrift. Springer. pp. 203–233. doi:10.1007/978-3-642-55566-4. ISBN 978-3-642-62442-1.
• 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.
• Werner, Tomas (2003). "Combinatorial constraints on multiple projections of set points". Proceedings Ninth IEEE International Conference on Computer Vision. pp. 1011–1016. doi:10.1109/ICCV.2003.1238459. ISBN 0-7695-1950-4. S2CID 6816538. Retrieved 26 November 2022.