Geometric continuity and compatibility conditions for 4-patch surfaces

TL;DR

This paper establishes universal geometric continuity conditions for 4-patch surfaces, ensuring tangent plane and curvature continuity at patch junctions, with applications to Bezier surfaces.

Contribution

It provides necessary and sufficient, parametrization-independent conditions for geometric regularity of 4-patch surfaces, extending previous results to general parametrizations.

Findings

Compatibility conditions are universal, independent of parametrization.

Conditions for tangent plane and curvature continuity are characterized.

Results are applied to Bezier patch representations.

Abstract

When considering regularity of surfaces, it is its geometry that is of interest. Thus, the concept of geometric regularity or geometric continuity of a specific order is a relevant concept. In this paper we discuss necessary and sufficient conditions for a 4-patch surface to be geometrically continuous of order one and two or, in other words, being tangent plane continuous and curvature continuous respectively. The focus is on the regularity at the point where the four patches meet and the compatibility conditions that must appear in this case. In this article the compatibility conditions are proved to be independent of the patch parametrization, i.e., the compatibility conditions are universal. In the end of the paper these results are applied to a specific parametrization such as Bezier representation in order to generalize a 4-patch surface result by Sarraga.