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I am working on figuring out the relationship between projective space (e.g. the unit sphere with antipodal points identified) and the perspective transform (e.g. the 2D image formed when light rays from a 3D environment pass through an ideal pinhole camera).

I have read that all conic sections (ellipses, hyperbolas, parabolas) are equivalent in projective geometry because they can all be interconverted via projective transforms. My understanding is that perspective transforms are a special case of projective transforms, and my question is this:

Which conic sections can be transformed into which others through merely perspective transforms?

(And what properties distinguish perspective transforms from more general projective transforms?)

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  • $\begingroup$ What is definition of a perspective transformation? $\endgroup$ – ziggurism Jan 1 '17 at 14:57
  • $\begingroup$ Here's an attempt: given a point ("center of perspectivity") and plane in R^3, define a map which sends each point in R^3 to the unique point at the intersection of the line joining that point to the center of perspectivity, and the plane. (or to a point @ infinity if there is no intersection). The perspective transforms comprise all such maps. $\endgroup$ – user326210 Jan 4 '17 at 7:06
  • $\begingroup$ so for example, a circle in the plane comes from the projection of a cone with the center of perspectivity at the apex of the cone. Projecting to a different plane from the same point can give you ellipses or hyberbolas, so I guess these are your projective transformations. However if you change the center of perspectivity and then project, you get 2D regions which are not even curves, so I guess these are not projective transformations. $\endgroup$ – ziggurism Jan 4 '17 at 14:16
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what properties distinguish perspective transforms from more general projective transforms?

Perspective transformations, at least in the sense you are talking about, where the three dimensional space is projected onto the two dimensional space, are different creatures from projective transformations, which are isomorphisms of projective spaces of equal dimension.

You get projective transformations naturally from perspective transformations, though, because you can usually fix attention on a plane in the picture (like a building facade, or the ground plane) and track how that is projected onto the camera image. Then you also have a projective transformation between any two cameras looking at the facade.

Which conic sections can be transformed into which others through merely perspective transforms?

I'm taking "conic" to mean that it is the set of null vectors of a nondegenerate metric. If you project, I think you can get "degenerate" conics from nondegenerate ones. I think one example is where you take a "cone" in projective 3 space, then project onto a projective plane containing a line lying on the cone.

It might be true that you can adjust everything by allowing degenerate conics. I can't totally remember at this point, but it might also depend on what the underlying field is. I have been thinking of this all through the lens of real projective spaces.

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