Smooth polar caps for locally quad-dominant meshes

A polar configuration is a node surrounded by m triangles. Polar configurations are common to cap off cylinders and spheres. When the triangles, interpreted as quadrilaterals with one edge collapsed, are surrounded by a quad-strip then the extended polar configuration qualifies as part of a locally quad dominant (lqd) mesh. Recent constructions, referred to as semi-structured splines, can use lqd meshes as control nets: multi-sided configurations that merge parameter directions are covered by G-spline; and T-junctions that transition from coarse and fine are covered by GT-splines. This paper complements existing semi-structured splines by providing the missing component for polar configurations. A spectrum of constructions of differing degree are introduced, tested and compared. Bi-2 C-1 splines are extended to polar configurations covered by C-1 surfaces consisting of (macro-)patches of degree as low as bi-2. Bi-3 C-2 splines are extended to polar configurations covered by surfaces that are C-2 except for a C-1 pole and consist of (macro-)patches of degree as low as bi-3. Crown Copyright (C) 2020 Published by Elsevier B.V. All rights reserved.