Rational function-based shape control for arbitrary topology surface subdivision

Abstract

This paper proposes a new non-stationary surface subdivision scheme driven by a rational control function. In regular regions, the scheme is defined by a tensor-product extension of a five-point univariate rule; around extraordinary vertices, we specify local update rules with a block-circulant structure. We prove C² smoothness in regular regions via first-order asymptotic equivalence with with the well-known Catmull--Clark scheme and establish tangent-plane continuity at extraordinary points by bounding the deviation of local blocks and invoking the spectral gap of the stationary limit. A parametric study shows how (a,b,μ) influences curvature and tension, yielding practical shape control without increasing computational complexity. We illustrate the method on standard test meshes and report indicative comparisons surface for representative parameter choices.