Parametrizations for triangular G k spline surfaces of low degree

Abstract

In this article, we present regularly parametrized G k free-form spline surfaces that extend box and half-box splines over regular triangular grids. The polynomial degree of these splines is max{4 k + 1, ⌈3k/2 + 1⌉r}, where r ∈ ℕ can be chosen arbitrarily and determines the flexibility at extraordinary points. The G k splines presented in this article depend crucially on low-degree (re-)parametrizations of piecewise polynomial hole fillings. The explicit construction of such parametrizations forms the core of this work and we present two classes of singular and regular parametrizations. Also, we show how to build box and half-box spline surfaces of arbitrarily high smoothness with holes bounded by only n patches, in principle.