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?)