Subdivisions of ring dupin cyclides using bézier curves with mass points

Abstract

Dupin cyclides are algebraic surfaces introduced for the first time in 1822 by the French mathematician Pierre-Charles Dupin. A Dupin cyclide can be defined as the envelope of a one-parameter family of oriented spheres, in two different ways. R. Martin is the first author who thought to use these surfaces in CAD/CAM and geometric modeling. The Minkowski-Lorentz space is a generalization of the space-time used in Einstein’s theory, equipped of the non-degenerate indefinite quadratic form QM (−→u ) = x2 + y2 + z2 − c2 t2 where (x, y, z) are the spacial components of the vector →−u and t is the time component of −→u and c is the constant of the speed of light. In this Minkowski-Lorentz space, a Dupin cyclide is the union of two conics on the unit pseudo-hypersphere, called the space of spheres, and a singular point of a Dupin cyclide is represented by an isotropic vector. Then, we model Dupin cyclides using rational quadratic Bézier curves with mass points. The subdivisions of a surface i.e. a Dupin cyclide, is equivalent to subdivide two curves of degree 2, independently, whereas in the 3D Euclidean space E3, the same work implies the subdivision of a rational quadratic Bézier surface and resolutions of systems of three linear equations. The first part of this work is to consider ring Dupin cyclides because the conics are circles which look like ellipses.