An Algorithm to Construct 3D Triangles with Circular Edges

Dupin cyclides are non-spherical algebraic surfaces of degree 4, discovered by the French mathematician Pierre-Charles Dupin at the beginning of the 19th century. A Dupin cyclide has a parametric equation and two implicit equations and circular lines of curvature. It can be defined as the image of a torus, a cone of revolution or a cylinder of revolution by an inversion. A torus has two families of circles : meridians and parallels. There is a third family of circles on a ring torus: Villarceau circles. As the image, by an inversion, of a circle is a circle or a straight line, there are three families of circles onto a Dupin cyclide too. The goal of this paper is to construct, onto a Dupin cyclide, 3D triangles with circular edges: a meridian arc, a parallel arc and a Villarceau circle arc.