Quadratic-attraction subdivision with contraction-ratio [formula omitted]

Abstract

Classic generalized subdivision, such as Catmull–Clark subdivision, as well as recent subdivision algorithms for high-quality surfaces, rely on slower convergence towards extraordinary points for mesh nodes surrounded by n>4 quadrilaterals. Slow convergence corresponds to a contraction-ratio of λ>0.5. To improve shape, prevent parameterization discordant with surface growth, or to improve convergence in isogeometric analysis near extraordinary points, a number of algorithms explicitly adjust λ by altering refinement rules. However, such tuning of λ has so far led to poorer surface quality, visible as uneven distribution or oscillation of highlight lines. The recent Quadratic-Attraction Subdivision (QAS) generates high-quality, bounded curvature surfaces based on a careful choice of quadratic expansion at the central point and, just like Catmull–Clark subdivision, creates the control points of the next subdivision ring by matrix multiplication. Unfortunately, QAS shares the contraction-ratio λCC>1/2 of Catmull–Clark subdivision when n>4. This shortcoming is finally remedied by the presented improvement QAS+ of QAS. For n=5,…,10, the convergence is made a uniform λ=12 as in tensor-product case and without sacrificing surface quality.