Convolution surfaces based on polygons for infinite and compact support kernels

We provide mathematical formulae to create 3D smooth shapes fleshing out a skeleton made of line segments and polygons. The boundary of the shape created is a level set of a convolution function. This latter is obtained through the integration of a kernel function along the skeleton. The technique has been proposed and developed for geometric modeling in computer graphics. Given the additivity of integration, the convolution function of a complex skeleton is the sum of the convolution functions for its simpler elements. Providing the closed form formulae for the convolution function generated by a polygon is the main contribution of the present paper. We apply Green’s theorem and improve on previous results in several ways. First we do not require the prior triangulation of the polygon. Then, we obtain formulae for complete families of kernels, either with infinite or compact supports. The families are indexed by an integer that controls sharpness and smoothness. Full generality is achieved through recurrence formulae. Last, but not least, the geometric computations needed, in the case of compact support kernels, are restricted to intersections of spheres with line segments,rather than intersections of spheres with triangles in previous works. keyword Green’s theorem; Polygonal Mesh; Implicit Surfaces; Convolution Surfaces; Geometric Modelling; Computer Graphics; Recurrences; Symbolic Integration. 1 in ria -0 05 36 84 0, v er si on 1 17 N ov 2 01 0 Author manuscript, published in "Graphical Models (2012)" DOI : 10.1016/j.gmod.2011.07.001