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 [math]\displaystyle{ \mathbb{R}_{*}^n }[/math] be the set of elements of [math]\displaystyle{ \mathbb{R}^n }[/math] excluding the origin.

1. Oriented projective line, [math]\displaystyle{ \mathbb{T}^1 }[/math]: [math]\displaystyle{ (x,w) \in \mathbb{R}^2_* }[/math], with the equivalence relation [math]\displaystyle{ (x,w)\sim(a x,a w)\, }[/math] for all [math]\displaystyle{ a\gt 0 }[/math].
2. Oriented projective plane, [math]\displaystyle{ \mathbb{T}^2 }[/math]: [math]\displaystyle{ (x,y,w) \in \mathbb{R}^3_* }[/math], with [math]\displaystyle{ (x,y,w)\sim(a x,a y,a w)\, }[/math] for all [math]\displaystyle{ a\gt 0 }[/math].

These spaces can be viewed as extensions of euclidean space. [math]\displaystyle{ \mathbb{T}^1 }[/math] can be viewed as the union of two copies of [math]\displaystyle{ \mathbb{R} }[/math], the sets (x,1) and (x,-1), plus two additional points at infinity, (1,0) and (-1,0). Likewise [math]\displaystyle{ \mathbb{T}^2 }[/math] can be viewed as two copies of [math]\displaystyle{ \mathbb{R}^2 }[/math], (x,y,1) and (x,y,-1), plus one copy of [math]\displaystyle{ \mathbb{T} }[/math] (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 [math]\displaystyle{ \mathbb T^n }[/math] is defined as

[math]\displaystyle{ \mathbb T^n=\{\{\lambda Z:\lambda\in\mathbb R_{\gt 0}\}:Z\in\mathbb R^{n+1}\setminus\{0\}\}=\{\mathbb R_{\gt 0}Z:Z\in\mathbb R^{n+1}\setminus\{0\}\}. }[/math]

Here, we use [math]\displaystyle{ \mathbb T }[/math] to stand for two-sided.

Distance in oriented real projective space

Distances between two points [math]\displaystyle{ p=(p_x,p_y,p_w) }[/math] and [math]\displaystyle{ q=(q_x,q_y,q_w) }[/math] in [math]\displaystyle{ \mathbb{T}^2 }[/math] can be defined as elements

[math]\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) }[/math]

in [math]\displaystyle{ \mathbb{T}^1 }[/math].

Oriented complex projective geometry

Let n be a nonnegative integer. The oriented complex projective space [math]\displaystyle{ {\mathbb{CP}}^n_{S^1} }[/math] is defined as

[math]\displaystyle{ {\mathbb{CP}}^n_{S^1}=\{\{\lambda Z:\lambda\in\mathbb R_{\gt 0}\}:Z\in\mathbb C^{n+1}\setminus\{0\}\}=\{\mathbb R_{\gt 0}Z:Z\in\mathbb C^{n+1}\setminus\{0\}\} }[/math]. Here, we write [math]\displaystyle{ S^1 }[/math] 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 et al.. 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. https://link.springer.com/book/10.1007/978-3-642-55566-4. 
• 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. https://ieeexplore.ieee.org/document/1238459. Retrieved 26 November 2022.