Top Qs
Timeline
Chat
Perspective

Oriented projective geometry

From Wikipedia, the free encyclopedia

Remove ads
Remove ads

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 be the set of elements of excluding the origin.

  1. Oriented projective line, : , with the equivalence relation for all .
  2. 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.
Remove ads

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

[4]

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]

Remove ads

Oriented complex projective geometry

Let n be a nonnegative integer. The oriented complex projective space is defined as

.[6] Here, we write to stand for the 1-sphere.
Remove ads

See also

Notes

Loading content...

References

Loading content...
Loading related searches...

Wikiwand - on

Seamless Wikipedia browsing. On steroids.

Remove ads