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 one parametric equation, two implicit equations, and a set of 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. A ring torus has an additional third family, called 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 defined as a meridian arc, a parallel arc, and an arc of a Villarceau circle.