A Quadratic $G^1$ Spline Approximation of the Sphere over Uniform Polyhedra

Abstract

In this paper, we study geometrically continuous quadratic splines over triangulations. While a rich variety of $C^1$ quadratic splines is available over planar domains, and such splines can also be constructed on the torus, the problem becomes significantly more challenging on more general surfaces. We first construct a $G^1$ spline over a regular spherical $n$-gon, subdivided into $3n$ triangles. Based on this construction, we obtain a quadratic $G^1$ spline approximation of the sphere induced by an arbitrary uniform polyhedron, where each $n$-gonal face is subdivided into $3n$ triangles. The construction uses only quadratic triangular patches and yields explicit control points depending on the geometry of the underlying polyhedron and one free parameter. We also analyze the resulting approximation quality and curvature behavior, and illustrate the construction on Platonic and Archimedean solids.