My interest in photography has led me off into some research into lens design. Why are camera lenses so darn complicated?
There are lots of issues, but the one I'd like to focus on is the shape of the lens. The shape of two objects rubbed against each other gradually becomes a sphere and a spherical hollow to match it. Most lens grinding is based on this, and so the surfaces of lens elements in a composite lens design are either flat, convex, or concave parts of a sphere.
Unfortunately, a sphere is not the perfect shape to focus light. That shape is a hyperbola. This has been known since the 17th century, and several great thinkers have tried to come up with ways to create hyperbolic surfaces on lenses.
A wonderful monograph on the subject is "Descartes And The Hyperbolic Quest", by D. Graham Burnett. I've linked to a page showing the solution of Christopher Wren, the great architect. I really like this image!
Now some of you might be thinking that this solution is cheating. Sure, you can create a hyperbolic surface if you've already got one handy, in the form of the hyperboloid. All you've done is push off the problem one step. But creating the hyperboloid is easy, as one of the other illustrations shows. If you have two skew lines, and rotate one around the other, you trace out the surface of a hyperboloid.
Even today, manufacturing costs dictate that most lens elements are spherical. Hyperbolic lenses are used in specialty applications such as copying machines. Aspherical elements are incorporated into sophisticated lens designs for cameras, but they are not to my knowledge hyperbolic.