High Quality Refinable G-splines for Locally Quad-dominant Meshes With T-gons

Abstract

Polyhedral modeling and re-meshing algorithms use T-junctions to add or remove feature lines in a quadrilateral mesh. In many ways this is akin to adaptive knot insertion in a tensor-product spline, but differs in that the designer or meshing algorithm does not necessarily protect the consistent combinatorial structure that is required to interpret the resulting quad-dominant mesh as the control net of a hierarchical spline - and so associate a smooth surface with the mesh as in the popular tensor-product spline paradigm. While G-splines for multi-sided holes or generalized subdivision can, in principle, convert quad-dominant meshes with T-junctions into smooth surfaces, they do not preserve the two preferred directions and so cause visible shape artifacts. Only recently have n-gons with T-junctions (T-gons) in unstructured quad-dominant meshes been recognized as a distinct challenge for generalized splines. This paper makes precise the notion of locally quad-dominant mesh as quad-meshes including t-nets, i.e. T-gons surrounded by quads; and presents the first high-quality G-spline construction that can use t-nets as control nets for spline surfaces suitable, e.g., for automobile outer surfaces. Remarkably, T-gons can be neighbors, separated by only one quad, both of T-gons and of points where many quads meet. A t-net surface cap consists of 16 polynomial pieces of degree (3,5) and is refinable in a way that is consistent with the surrounding surface. An alternative, everywhere bi-3 cap is not formally smooth, but achieves the same high-quality highlight line distribution.