Lesson plan
Oriented projective geometry
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 R 3 , 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 \mathbb{R}_{*}^n be the set of elements of \mathbb{R}^n excluding the origin. #Oriented projective line, \mathbb{T}^1 : (x,w) \in \mathbb{R}^2_* , with the equivalence relation (x,w)\sim(a x,a w)\, for all a>0 . #Oriented projective plane, \mathbb{T}^2 : (x,y,w) \in \mathbb{R}^3_* , with (x,y,w)\sim(a x,a y,a w)\, for all a>0 . These spaces can be viewed as extensions of euclidean space. \mathbb{T}^1 can be viewed as the union of two copies of \mathbb{R} , the sets (x,1) and (x,-1), plus two additional points at infinity, (1,0) and (-1,0). Likewise \mathbb{T}^2 can be viewed as two copies of \mathbb{R}^2 , (x,y,1) and (x,y,-1), plus one copy of \mathbb{T} (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 2 +y 2 +w 2 =1.
Article summary
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 R 3 , 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 \mathbb{R}_{*}^n be the set of elements of \mathbb{R}^n excluding the origin. #Oriented projective line, \mathbb{T}^1 : (x,w) \in \mathbb{R}^2_* , with the equivalence relation (x,w)\sim(a x,a w)\, for all a>0 . #Oriented projective plane, \mathbb{T}^2 : (x,y,w) \in \mathbb{R}^3_* , with (x,y,w)\sim(a x,a y,a w)\, for all a>0 . These spaces can be viewed as extensions of euclidean space. \mathbb{T}^1 can be viewed as the union of two copies of \mathbb{R} , the sets (x,1) and (x,-1), plus two additional points at infinity, (1,0) and (-1,0). Likewise \mathbb{T}^2 can be viewed as two copies of \mathbb{R}^2 , (x,y,1) and (x,y,-1), plus one copy of \mathbb{T} (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 2 +y 2 +w 2 =1.
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Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
- Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
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