Quadratic-attraction subdivision with contraction-ratio λ=12

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 $\lambda >0.5$. 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 $\lambda$ by altering refinement rules. However, such tuning of $\lambda$ 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. But QAS shares the contraction-ratio ${\lambda }_{CC}>1/2$ of Catmull–Clark subdivision when $n>4$. For $n=5,\dots ,10$, QAS${}_{+}$ improves the convergence to the uniform $\lambda =\frac{1}{2}$ of binary domain refinement and without sacrificing surface quality compared to QAS.