|
|
| Line 1: |
Line 1: |
| '''Oriented projective geometry''' is an [[orientability|oriented]] version of real [[projective geometry]].
| | The writer is known as Irwin Wunder but it's not the most masucline name out there. My day job is a meter reader. One of the things she enjoys most is to read comics and she'll be starting some thing else alongside with it. California is our birth place.<br><br>Take a look at my webpage; [http://dns125.dolzer.at/?q=node/696 http://dns125.dolzer.at/?q=node/696] |
| | |
| Whereas the [[real projective plane]] describes the set of all unoriented lines through the origin in '''R'''<sup>3</sup>, 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>\mathbf{R}_{*}^n</math> be the set of elements of <math>\mathbf{R}^n</math> excluding the origin.
| |
| #'''Oriented projective line''', <math>\mathbf{T}^1</math>: <math>(x,w) \in \mathbf{R}^2_*</math>, with the [[equivalence relation]] <math>(x,w)\sim(a x,a w)\,</math> for all <math>a>0</math>.
| |
| #'''Oriented projective plane''', <math>\mathbf{T}^2</math>: <math>(x,y,w) \in \mathbf{R}^3_*</math>, with <math>(x,y,w)\sim(a x,a y,a w)\,</math> for all <math>a>0</math>.
| |
| | |
| These spaces can be viewed as extensions of [[euclidean space]]. <math>\mathbf{T}^1</math> can be viewed as the union of two copies of <math>\mathbf{R}</math>, the sets (''x'',1) and (''x'',-1), plus two additional points at infinity, (1,0) and (-1,0). Likewise <math>\mathbf{T}^2</math> can be view two copies of <math>\mathbf{R}^2</math>, (''x'',''y'',1) and (''x'',''y'',-1), plus one copy of <math>\mathbf{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
| |
| | |
| :''x''<sup>2</sup>+''y''<sup>2</sup>+''z''<sup>2</sup>=1.
| |
| | |
| Distances between two points
| |
| | |
| :<math>p=(p_x,p_y,p_w)</math>
| |
| | |
| and
| |
| | |
| :<math>q=(q_x,q_y,q_w)</math>
| |
| | |
| in
| |
| | |
| :<math>\mathbf{T}^2</math>
| |
| | |
| can be defined as elements in <math>\mathbf{T}^1</math>
| |
| :<math>((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>
| |
| | |
| ==References==
| |
| * {{cite book
| |
| | last = Stolfi
| |
| | first = Jorge
| |
| | title = Oriented Projective Geometry
| |
| | publisher = [[Academic Press]]
| |
| | date = 1991
| |
| | isbn = 978-0-12-672025-9 }}<br />From original [[Stanford]] Ph.D. dissertation, ''Primitives for Computational Geometry'', available as [http://www.hpl.hp.com/techreports/Compaq-DEC/SRC-RR-36.pdf].
| |
| * 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
| |
| | last = Ghali
| |
| | first = Sherif
| |
| | title = Introduction to Geometric Computing
| |
| | publisher = [[Springer Science+Business Media|Springer]]
| |
| | date = 2008
| |
| | isbn = 978-1-84800-114-5 }}<br />Nice introduction to oriented projective geometry in chapters 14 and 15. More at authors web-site. [http://www.dgp.toronto.edu/~ghali/ Sherif Ghali].
| |
| | |
| {{geometry-stub}}
| |
| [[Category:Projective geometry]]
| |
The writer is known as Irwin Wunder but it's not the most masucline name out there. My day job is a meter reader. One of the things she enjoys most is to read comics and she'll be starting some thing else alongside with it. California is our birth place.
Take a look at my webpage; http://dns125.dolzer.at/?q=node/696