This article needs attention from an expert in Mathematics or computer science. The specific problem is: explain or correct the phrase " (x,y,0)", and the distance formula seems incorrect (missing a square root? (cf. Section 17.4 of Stolfi)) and could be better written. See the
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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 be the set of elements of excluding the origin.
These spaces can be viewed as extensions of
euclidean space. can be viewed as the union of two copies of , the sets (x,1) and (x,-1), plus two additional points at infinity, (1,0) and (-1,0). Likewise can be viewed as two copies of , (x,y,1) and (x,y,-1), plus one copy of (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 is defined as
Stolfi, Jorge (1991). Oriented Projective Geometry.
Academic Press.
ISBN978-0-12-672025-9. From original
Stanford Ph.D. dissertation, Primitives for Computational Geometry, available as
[1].
Ghali, Sherif (2008). Introduction to Geometric Computing.
Springer.
ISBN978-1-84800-114-5. Nice introduction to oriented projective geometry in chapters 14 and 15. More at author's website.
Sherif Ghali.
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.