Oriented projective geometry

{{Expert needed|mathematics|ex2=computer science|talk=Section title goes here|reason=explain or correct the phrase " $\mathbb{T}$ (x,y,0)", and the distance formula seems incorrect (missing a square root? (cf. Section 17.4 of Stolfi)) and could be better written|date=November 2022}}

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.

Oriented projective line, $\mathbb{T}^1$ : $(x,w) \in \mathbb{R}^2_*$ , with the equivalence relation $(x,w)\sim(a x,a w)\,$ for all $a>0$ .
Oriented projective plane, $\mathbb{T}^2$ : $(x,y,w) \in \mathbb{R}^3_*$ , 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. $\mathbb{T}^1$ 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{{sfn|Stolfi|1991|p=2}}) oriented (real) projective space or (canonical{{sfn|Stolfi|1991|p=13}}) two-sided projective{{sfn|Werner|2003}} space $\mathbb T^n$ is defined as

: $\mathbb T^n=\{\{\lambda Z:\lambda\in\mathbb R_{>0}\}:Z\in\mathbb R^{n+1}\setminus\{0\}\}=\{\mathbb R_{>0}Z:Z\in\mathbb R^{n+1}\setminus\{0\}\}.$ {{sfn|Yamaguchi|2002|pp=33–34, Definition 4.1}}

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 $\mathbb{T}^1$ .{{sfn|Stolfi|1991|loc=§17.4}}

Oriented complex projective geometry

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

: ${\mathbb{CP}}^n_{S^1}=\{\{\lambda Z:\lambda\in\mathbb R_{>0}\}:Z\in\mathbb C^{n+1}\setminus\{0\}\}=\{\mathbb R_{>0}Z:Z\in\mathbb C^{n+1}\setminus\{0\}\}$ .{{sfn|Below|2003}} Here, we write $S^1$ to stand for the 1-sphere.

Notes

References

{{cite book

| last = Stolfi

| first = Jorge

| title = Oriented Projective Geometry

| publisher = Academic Press

| date = 1991

| isbn = 978-0-12-672025-9 }}
From original Stanford Ph.D. dissertation, Primitives for Computational Geometry, available as [http://www.hpl.hp.com/techreports/Compaq-DEC/SRC-RR-36.pdf].

{{cite book

| last = Ghali

| first = Sherif

| title = Introduction to Geometric Computing

| publisher = Springer

| date = 2008

| isbn = 978-1-84800-114-5 }}
Nice introduction to oriented projective geometry in chapters 14 and 15. More at author's website. [http://www.dgp.toronto.edu/~ghali/ Sherif Ghali].

{{cite book |last1=Yamaguchi |first1=Fujio |title=Computer-aided Geometric Design: A Totally Four-dimensional Approach |date=2002 |publisher=Springer |isbn=978-4-431-68007-9}}
{{cite book |last1=Below |first1=Alexander |last2=Krummeck |first2=Vanessa |last3=Richter-Gebert |first3=Jurgen |editor1-last=Aronov |editor1-first=Boris |editor1-link=Boris Aronov |editor2-last=Basu |editor2-first=Saugata |editor3-last=Pach |editor3-first=Janos |editor3-link=Janos Pach |editor4-last=Sharir |editor4-first=Micha |editor4-link=Micha Sharir |title=Discrete and Computational Geometry: The Goodman–Pollack Festschrift |chapter=Complex matroids: phirotopes and their realizations in rank 2 |date=2003 |publisher=Springer |isbn=978-3-642-62442-1 |pages=203–233 |doi=10.1007/978-3-642-55566-4 |url=https://link.springer.com/book/10.1007/978-3-642-55566-4 |ref={{harvid|Below|2003}}}}
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}}

Category:Projective geometry