Some comments on quadric surfaces

We are interested in surfaces in three-dimensional space satisfied by equations of the form

ax²+by²+cz²+dxy+exz+fyz+gx+hy+iz+j=0

We are most interested in forms which can be, via translation and rotation, be transformed into quadratic polynomials of the form

which are quadratic in at least two of the variables. Such forms result in surfaces known as quadric surfaces, and there are six types. The other forms result in cylinders, planes, or degenerate equations with no solutions. Below are some comments on quadric surfaces.

• Cones, hyperpoloids of one sheet, and hyperboloids of two sheets

The generic forms of the equations of these three types of surfaces can be written as follows:



Note that these are identical except for the constant on the right-hand side. Thus we can think of all of these surfaces are arising from equations of the form



where k is a real constant, and the surface is a cone if k=0, a hyperboloid of one sheet if k is positive, and a hyperboloid of two sheets if k is negative.

Here is an animation showing this spectrum of surfaces resulting from this equation form. In the animation, the constant k varies from positive to negative and back. The cone is seen when k hits zero.