# Which conic sections are related through perspective transforms?

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?)

• What is definition of a perspective transformation? – ziggurism Jan 1 '17 at 14:57
• 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. – user326210 Jan 4 '17 at 7:06
• 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. – ziggurism Jan 4 '17 at 14:16