A Hermite interpolatory subdivision scheme for $C^2$-quintics on the Powell-Sabin 12-split

TL;DR

This paper introduces a Hermite subdivision scheme for efficiently constructing visually appealing $C^2$-quintic spline surfaces on Powell-Sabin 12-splits, enabling quick evaluation and high-quality rendering.

Contribution

It presents a novel nodal macro-element for quintic splines on the Powell-Sabin 12-split and derives a Hermite subdivision scheme for fast, high-quality spline evaluation.

Findings

The scheme allows quick evaluation of $C^2$-quintic splines.

Visualizations with Phong shading are improved after few refinements.

Implementation in Sage demonstrates practical efficiency.

Abstract

In order to construct a $C^1$-quadratic spline over an arbitrary triangulation, one can split each triangle into 12 subtriangles, resulting in a finer triangulation known as the Powell-Sabin 12-split. It has been shown previously that the corresponding spline surface can be plotted quickly by means of a Hermite subdivision scheme. In this paper we introduce a nodal macro-element on the 12-split for the space of quintic splines that are locally $C^3$ and globally $C^2$. For quickly evaluating any such spline, a Hermite subdivision scheme is derived, implemented, and tested in the computer algebra system Sage. Using the available first derivatives for Phong shading, visually appealing plots can be generated after just a couple of refinements.