Lens Design ?

You may need to find a technical tome on lens design, or perhaps search a patent database for "hyperboloid" and "lens". Hyperboloidal *mirrors* are used in astrophotography in the Ritchey-Chrétien telescopes used in professional and high-end amateur observatories, and in a catadioptric design known as a hyperbolic astrograph. Camera manufacturers often claim aspheric lens elements but aren't generally forthcoming as to the shape used; I'd guess most of them are paraboloids to correct spherical aberration.

a hyperbola is a U shape right? cut a clear, flat plastic bottle in half, so you have a U shape and stare through it? what happens? the image gets stretched out into an oval with the clearest part being in the center or at the bottom of the U where the curve is.