Dupin Cyclides

A Dupin cyclide is the envelope of a one-parameter family of spheres tangent to three fixed spheres. By studying the relation between a Dupin cyclide and a torus of revolution, partly from an inversive geometry approach, we prove that the definition of the first is equivalent to being the conformal image of a torus of revolution. Here we define a torus of revolution such that all standard tori fall within the definition. Furthermore, we look into the curvature lines of a Dupin cyclide, where we use amongst others the theorems of Joachimsthal and Meusnier, and prove that the two previous characterizations of a Dupin cyclide are equivalent with being a surface all whose lines of curvature are circles.