Do all smoothly curved pieces of transparent material have a focal point? I understand that a lens by definition has a focal point but do all pieces of glass with smoothly curved surfaces have a single point at which all parallel rays of light passing through it in a certain direction theoretically meet? I'm sure running a bunch of shapes through ray tracing programs could provide a satisfactory answer but I'm wondering if there's a formal mathematical proof? I've looked around and I can't seem to find anything on the matter.
 A: No, even a lens don,t have a single focal point. they have a range of points in their principal  axis . but for a very very large radius of curvature of lens compared to compared to its aperture, these range gets smaller and smaller eventually it tends to become a single point it is what we normal call a focal point. with irregular surfaces having many radius of curvatures, they have many ranges of focal point (a lens with two radius of curvature have 2 range of focal points and a mirror with one radius of curvature have one range of focal points)
A: Read the paragraph between Fig 69 and Fig 70 on this page. Similar statements apply for lenses. The only kind of mirror/lens that focuses parallel rays exactly onto its focal point is a parabolic mirror/lens.
A: Yes, if this transparent material is working like a convex lens, create always a focal point. http://www.passmyexams.co.uk/GCSE/physics/images/convex_lens.jpg
Not all the materials create a focal point. For example, if your transparent material is more or less similar to other type of lenses (like concaves) don´t allow to spread the light in one point
