Smooth Ternary Subdivision Surfaces with Bounded Curvature

Abstract

Binary subdivision schemes, such as Cat mull-Clark scheme or Loop scheme, fail to characterize local shape near the irregular vertex. Address the problem, this paper proposes a smooth ternary subdivision scheme for quadrilateral meshes with bounded curvature. The scheme is based on a 1-9 splitting operator. The subdivision rules for regular vertices are derived from Bi-cubic B-spline surface and the rules for irregular vertices are established through the Fourier analysis of the regular case. By analyzing the eigen structures and characteristic maps, the proposed subdivision scheme can produces C <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> continuous limit surfaces for regular meshes while achieves G <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> continuity at irregular vertices. Compared with typical binary subdivision schemes, the proposed scheme has bounded curvatures and the fast convergence speed. Furthermore, it maintains arbitrary topological quadrilateral meshes.