Manifold splines with a single extraordinary point

作者：Gu Xianfeng; He Ying^*; Jin Miao; Luo Feng; Qin Hong; Yau Shing Tung

来源：CAD Computer Aided Design, 2008, 40(6): 676-690.

DOI：10.1016/j.cad.2008.01.008

• manifold splines
• affine structure
• discrete Ricci flow
• extraordinary points
• metric
• differential geometry

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 flow any planar domain to all arbitrary topology. As a result, manifold splines are to systematically extend 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., vertical bar 2g - 2 vertical bar for any gesus-g surface), where the analysis of surface behaviors such as continuity remains extremely difficult. The unique theoretical contribution of this paper is that we device new mathematical tools so that manifold splines call now be constructed with only one singular point, reaching their theoretic lower bound of similarity 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 atone point. Then, the Metric naturally induces all 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.

• 出版日期2008-6
• 单位南阳理工学院