Geometric Modeling

Abstract

In this chapter, we apply computational conformal geometry to construct manifold splines. We show that manifold splines afford a general theoretical and computational framework for modeling geometrically complicated surfaces of arbitrary topology. The technical challenge is how to extend polynomial-centric splines defined over open, planar domains to that over any manifold setting. Our solution is an affine structure for any manifold surface, serving as a parametric domain, so that piecewise spline functions can be naturally and elegantly blended to represent geometric shapes of arbitrary topology. Built upon our prior efforts, the primary foci are to broaden and strengthen the theoretical foundation as well as to devise practical algorithms for manifold splines. In particular, we advocate several novel mathematical tools to compute affine atlas for any domain manifold. At the theoretic level, we show that the lower bound of the number of singularities for surfaces with non-zero Euler number is one. We make use of discrete Ricci flow to actually reach this lower bound for manifold spline construction. At the practical level, we further relax this rather strict requirement by allowing users to control the number and positions of singular points. As a result, we construct polycube splines as a novel variant of manifold splines. Our computational tool is polycube maps, which can reduce both the area and angle distortion in the affine atlas. In order to demonstrate the general feasibility and efficiency of manifold splines, we design algorithms to extend various planar splines, such as triangular B-splines, Powell-Sabin splines, and T-splines to the manifold settings. We also highlight their modeling advantages and potentials in shape representation and analysis/synthesis through a wide array of experiments.