Dupin [1822] determined which surfaces M embedded in Euclidean threespace ℝ33 can be obtained as the envelope of the family of spheres tangent to three fixed spheres. These surfaces, known as the cyclides of Dupin, can all be constructed by inverting a torus of revolution, a circular cylinder or a circular cone in a sphere. The cyclides of Dupin were studied extensively in the nineteenth century (see, for example, [Cayley 1873; Liouville 1847; Maxwell 1867]). They have several other important characterizations. They are the only surfaces M in ℝ3 whose focal set consists of two curves, which must, in fact, be a pair of focal conies. This is equivalent to requiring that M have two distinct principal curvatures at every point, each of which is constant along each of its corresponding lines of curvature. It is also equivalent to the condition that all lines of curvature in both families are circles or straight lines.