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

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

These spaces can be viewed as extensions of euclidean space. ${\displaystyle \mathbb {T} ^{1}}$ can be viewed as the union of two copies of ${\displaystyle \mathbb {R} }$, the sets (x,1) and (x,-1), plus two additional points at infinity, (1,0) and (-1,0). Likewise ${\displaystyle \mathbb {T} ^{2}}$ can be viewed as two copies of ${\displaystyle \mathbb {R} ^{2}}$, (x,y,1) and (x,y,-1), plus one copy of ${\displaystyle \mathbb {T} }$ (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 ^[1] ) oriented (real) projective space or (canonical ^[2] ) two-sided projective ^[3] space${\displaystyle \mathbb {T} ^{n}}$ is defined as

${\displaystyle \mathbb {T} ^{n}=\{\{\lambda Z:\lambda \in \mathbb {R} _{>0}\}:Z\in \mathbb {R} ^{n+1}\setminus \{0\}\}=\{\mathbb {R} _{>0}Z:Z\in \mathbb {R} ^{n+1}\setminus \{0\}\}.}$ ^[4]

Here, we use ${\displaystyle \mathbb {T} }$ to stand for two-sided.

Distance in oriented real projective space

Distances between two points ${\displaystyle p=(p_{x},p_{y},p_{w})}$ and ${\displaystyle q=(q_{x},q_{y},q_{w})}$ in ${\displaystyle \mathbb {T} ^{2}}$ can be defined as elements

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

in ${\displaystyle \mathbb {T} ^{1}}$. ^[5]

Oriented complex projective geometry

See also: Complex projective space

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

${\displaystyle {\mathbb {CP} }_{S^{1}}^{n}=\{\{\lambda Z:\lambda \in \mathbb {R} _{>0}\}:Z\in \mathbb {C} ^{n+1}\setminus \{0\}\}=\{\mathbb {R} _{>0}Z:Z\in \mathbb {C} ^{n+1}\setminus \{0\}\}}$. ^[6] Here, we write ${\displaystyle S^{1}}$ to stand for the 1-sphere.

See also

• Variational analysis

Notes

1. Stolfi 1991, p. 2.
2. Stolfi 1991, p. 13.
3. Werner 2003.
4. Yamaguchi 2002, pp. 33–34, Definition 4.1.
5. Stolfi 1991, §17.4.
6. Below 2003.

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 Archived 2021-06-11 at the Wayback Machine .
• 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" (PDF). Proceedings Ninth IEEE International Conference on Computer Vision. pp. 1011–1016. doi:10.1109/ICCV.2003.1238459. ISBN   0-7695-1950-4.