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Oriented projective geometry
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Oriented projective geometry is an oriented version of real projective geometry.
![]() | This article includes a list of references, related reading, or external links, but its sources remain unclear because it lacks inline citations. (November 2022) |
![]() | 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 talk page for details. (November 2022) |
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.
- Oriented projective line, : , with the equivalence relation for all .
- Oriented projective plane, : , with for all .
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.
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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
Here, we use to stand for two-sided.
Distance in oriented real projective space
Distances between two points and in can be defined as elements
in .[5]
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Oriented complex projective geometry
Let n be a nonnegative integer. The oriented complex projective space is defined as
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See also
Notes
References
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