Dupin cyclides

Abstract

Introduction: We will study the problem of finding a Dupin cyclide given three contact conditions. A contact condition is a point m in R3 and an orientation at that point, i.e., a vector with initial point m, this vector can be seen as the normal vector of an oriented plane through m. A Dupin cyclide that, historically, is defined as the envelope of a family of spheres which are simultaneously tangent to three given spheres, can be defined in the context of the projective five-dimensional space as 2-plane section of a quadric in this projective space. The points of this quadric correspond to oriented spheres in 3D.