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

1. Oriented projective line, $\mathbf{T}^1$: $(x,w) \in \mathbf{R}^2_*$, with the equivalence relation $(x,w)\sim(a x,a w)\,$ for all a > 0.
2. Oriented projective plane, $\mathbf{T}^2$: $(x,y,w) \in \mathbf{R}^3_*$, 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. $\mathbf{T}^1$ can be viewed as the union of two copies of $\mathbf{R}$, the sets (x,1) and (x,-1), plus two additional points at infinity, (1,0) and (-1,0). Likewise $\mathbf{T}^2$ can be view two copies of $\mathbf{R}^2$, (x,y,1) and (x,y,-1), plus one copy of $\mathbf{T}$ (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+z²=1.

Distances between two points

p = (p_x,p_y,p_w)

and

q = (q_x,q_y,q_w)

in

$\mathbf{T}^2$

can be defined as elements in $\mathbf{T}^1$

$((p_x q_w-q_x p_w)^2+(p_y q_w-q_y p_w)^2,\mathrm{sign}(p_w q_w)(p_w q_w)^2)\,.$

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 DEC SRC Research Report 36.
• 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.
• Ghali, Sherif (2008). Introduction to Geometric Computing. Springer. ISBN 978-1848001145. 
  Nice introduction to oriented projective geometry in chapters 14 and 15. More at authors web-site. Sherif Ghali.

This geometry-related article is a stub. You can help Wikipedia by expanding it.v · Categories: Geometry stubs Projective geometry Wikimedia Foundation. 2010. Игры ⚽ Поможем написать реферат William E. Peterson It Takes a Thief Look at other dictionaries: Oriented matroid — theory allows a combinatorial approach to the max flow min cut theorem. A network with the value of flow equal to the capacity of an s t cut An oriented matroid is a mathematical structure that abstracts the properties of directed graphs and of… …   Wikipedia Flat (geometry) — In geometry, a flat is a subset of n dimensional space that is congruent to a Euclidean space of lower dimension. The flats in two dimensional space are points and lines, and the flats in three dimensional space are points, lines, and planes.In n …   Wikipedia Inversive geometry — Not to be confused with Inversive ring geometry. In geometry, inversive geometry is the study of those properties of figures that are preserved by a generalization of a type of transformation of the Euclidean plane, called inversion. These… …   Wikipedia Lie sphere geometry — is a geometrical theory of planar or spatial geometry in which the fundamental concept is the circle or sphere. It was introduced by Sophus Lie in the nineteenth century. [The definitive modern textbook on Lie sphere geometry is Harvnb|Cecil|1992 …   Wikipedia Differential geometry of surfaces — Carl Friedrich Gauss in 1828 In mathematics, the differential geometry of surfaces deals with smooth surfaces with various additional structures, most often, a Riemannian metric. Surfaces have been extensively studied from various perspectives:… …   Wikipedia Discrete geometry — A collection of circles and the corresponding unit disk graph Combinatorial geometry redirects here. The term combinatorial geometry is also used in the theory of matroids to refer to a simple matroid, especially in older texts. Discrete geometry …   Wikipedia Riemannian geometry — Elliptic geometry is also sometimes called Riemannian geometry. Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric , i.e. with an inner product on the tangent… …   Wikipedia Dowling geometry — In combinatorial mathematics, a Dowling geometry, named after Thomas A. Dowling, is a matroid associated with a group. There is a Dowling geometry of each rank for each group. If the rank is at least 3, the Dowling geometry uniquely determines… …   Wikipedia List of mathematics articles (O) — NOTOC O O minimal theory O Nan group O(n) Obelus Oberwolfach Prize Object of the mind Object theory Oblate spheroid Oblate spheroidal coordinates Oblique projection Oblique reflection Observability Observability Gramian Observable subgroup… …   Wikipedia Plücker coordinates — In geometry, Plücker coordinates, introduced by Julius Plücker in the 19th century, are a way to assign six homogenous coordinates to each line in projective 3 space, P 3. Because they satisfy a quadratic constraint, they establish a one to one… …   Wikipedia 18+ © Academic, 2000-2024 Contact us: Technical Support, Advertising Dictionaries export, created on PHP, Joomla, Drupal, WordPress, MODx. Mark and share Search through all dictionaries Translate… Search Internet Share the article and excerpts Direct link … Do a right-click on the link above and select “Copy Link”