Geometric continuity and compatibility conditions for 4-patch surfaces
Provides universal compatibility conditions for 4-patch surfaces, which is an incremental theoretical contribution for geometric modeling.
This paper derives necessary and sufficient conditions for geometric continuity of order one and two (tangent plane and curvature continuity) at the central point of a 4-patch surface, proving that the compatibility conditions are independent of patch parametrization. The results generalize a previous result by Sarraga for Bézier surfaces.
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.