The paper deals in the Computer-Aided Design or Computer-Aided Manufacturing domain with the Dupin cyclides as well as the Bézier curves. It shows that the same algorithms can be used either for subdivisions of ring Dupin cyclides or Bézier curves. The Bézier curves are described with mass points here. The Dupin cyclides are considered in the Minkowski-Lorentz space. This makes a Dupin cyclide as the union of two conics on the unit pseudo-hypersphere, called the space of spheres. And the conics are quadratic Bézier curves modelled by mass points. The subdivision of any Dupin cyclide, is equivalent to subdivide two curves of degree 2, independently, whereas in the 3D Euclidean space, 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 bounded circles which look like ellipses.