Summary: Boolean Operations with Implicit and
Parametric Representation of
Primitives Using R-Functions
Yohan D. Fougerolle, Andrei Gribok, Sebti Foufou,
FreŽdeŽric Truchetet, Member, IEEE, and Mongi A. Abidi, Member, IEEE
Abstract--We present a new and efficient algorithm to accurately polygonize an implicit surface generated by multiple Boolean
operations with globally deformed primitives. Our algorithm is special in the sense that it can be applied to objects with both an implicit
and a parametric representation, such as superquadrics, supershapes, and Dupin cyclides. The input is a Constructive Solid Geometry
tree (CSG tree) that contains the Boolean operations, the parameters of the primitives, and the global deformations. At each node of
the CSG tree, the implicit formulations of the subtrees are used to quickly determine the parts to be transmitted to the parent node,
while the primitives' parametric definition are used to refine an intermediary mesh around the intersection curves. The output is both an
implicit equation and a mesh representing its solution. For the resulting object, an implicit equation with guaranteed differential
properties is obtained by simple combinations of the primitives' implicit equations using R-functions. Depending on the chosen
R-function, this equation is continuous and can be differentiable everywhere. The primitives' parametric representations are used to
directly polygonize the resulting surface by generating vertices that belong exactly to the zero-set of the resulting implicit equation. The
proposed approach has many potential applications, ranging from mechanical engineering to shape recognition and data compression.
Examples of complex objects are presented and commented on to show the potential of our approach for shape modeling.
Index Terms--Computational geometry and object modeling, constructive solid geometry, object representation, volume visualization,
R-functions, superquadrics, supershapes, Dupin cyclides.