Manifold Splineswith Single Extraordinary Point
Computer Aided Design (CAD 2008)
Xianfeng Gu, Ying He, Miao Jin, Feng Luo, Hong Qin and Shing-Tung Yau
This paper develops a novel computational technique to define and construct manifold splines with only one singular point by employing
the rigorous mathematical theory of Ricci flow. The central idea and new computational paradigm of manifold splines are to systematically
extend the algorithmic pipeline of spline surface construction from any planar domain to arbitrary topology. As a result, manifold splines
can unify planar spline representations as their special cases. Despite its earlier success, the existing manifold spline framework is plagued
by the topology-dependent, large number of singular points (i.e., |2g−2| for any genus-g surface), where the analysis of surface behaviors
such as continuity remains extremely difficult. The unique theoretical contribution of this paper is that we devise new mathematical tools so
that manifold splines can now be constructed with only one singular point, reaching their theoretic lower bound of singularity for real-world
applications. Our new algorithm is founded upon the concept of discrete Ricci flow and associated techniques. First, Ricci flow is employed
to compute a special metric of any manifold domain (serving as a parametric domain for manifold splines), such that the metric becomes
flat everywhere except at one point. Then, the metric naturally induces an affine atlas covering the entire manifold except this singular point.
Finally, manifold splines are defined over this affine atlas. The Ricci flow method is theoretically sound, and practically simple and efficient.
We conduct various shape experiments and our new theoretical and algorithmic results alleviate the modeling difficulty of manifold splines,
and hence, promote the widespread use of manifold splines in surface and solid modeling, geometric design, and reverse engineering.