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| '''Oriented projective geometry''' is an [[orientability|oriented]] version of real [[projective geometry]].
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| 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.
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| 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.
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| #'''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>.
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| #'''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>.
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| 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).
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| An alternative way to view the spaces is as points on the circle or sphere, given by the points (''x'',''y'',''w'') with
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| :''x''<sup>2</sup>+''y''<sup>2</sup>+''z''<sup>2</sup>=1.
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| Distances between two points
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| :<math>p=(p_x,p_y,p_w)</math>
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| and
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| :<math>q=(q_x,q_y,q_w)</math>
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| in
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| :<math>\mathbf{T}^2</math>
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| can be defined as elements in <math>\mathbf{T}^1</math>
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| :<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>
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| ==References==
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| * {{cite book
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| | last = Stolfi
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| | first = Jorge
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| | title = Oriented Projective Geometry
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| | publisher = [[Academic Press]]
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| | date = 1991
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| | 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].
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| * 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.
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| * {{cite book
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| | last = Ghali
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| | first = Sherif
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| | title = Introduction to Geometric Computing
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| | publisher = [[Springer Science+Business Media|Springer]]
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| | date = 2008
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| | 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].
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| {{geometry-stub}}
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| [[Category:Projective geometry]]
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