This paper shows the same algorithm is used for subdivisions of Dupin cyclides with singular points and quadratic Bézier curves passing through infinity. The mass points are usefull for any quadratic Bézier representation of a parabola or an hyperbola arc. The mass points are mixing weighted points and pure vectors. Any Dupin cyclide is considered in the Minkowski-Lorentz space. In that space, the Dupin cyclide is defined by the union of two conics laying on the unit pseudo-hypersphere, called the space of spheres. The subdivision of any Dupin cyclide, is equivalent to subdivide two Bézier curves of degree 2 with mass points, independently. The use of these two curves eases the subdivision of a Dupin cyclide patch or triangle.