Quadric Surfaces

A surface described by a second-degree equation is far more constrained than one might expect: there are only a handful of shapes. This chapter works through them one by one — the sphere, the ellipsoid, the paraboloids, the hyperboloids, the cone and the cylinders — and for each one gives the standard equation, the geometric properties and the sections that identify it.

The reason there are so few shapes is that the type of a quadric surface is decided by the signs of the eigenvalues of its quadratic form. The ellipse family has all three eigenvalues of the same sign, the hyperbola family has two of one sign and one of another, the parabola family has exactly one zero eigenvalue, and the cylinder family is the degenerate case. Once you have this classification in mind, the standard equations stop being a list to be memorised and become a consequence you can derive.

The last section gives the methods of classification — the eigenvalue test, the invariant test and the method of direct observation — so that you can work out the type of an arbitrary second-degree equation rather than only recognising the ones you have seen before.