Implicit and Parametric Representations of Three Blobs

A surface can be represented mainly using two mathematical models by using either a parametric equation or an implicit equation. The intuitive use and facility to represent just a part of surface make the first type, known as parametric, very popular, especially in industrial applications (Bezier, B-Splines, NURBS). The second type of surface, known as implicit, is employed in applications of ray tracing, surface illumination, intersection computation or determination of the interior of objects delimited by their edges. Thus, it is essential for geometrical modeling to have mathematical objects having at the same time a parametric equation and if possible an algebraic implicit equation. It is interesting to obtain an implicit equation of degree relatively low in order to limit the size of calculations. We already solved the problem for the fusion of 2 Blobs and found an analytical solution allowing passing easily from an implicit representation to a parametric representation. We propose here to extend these results to the case of three blobs with the restriction that the 3 centers are aligned on the same axis. In this paper we initially point out our method for two blobs, and then we will present our extension for three blobs.