Geometric Properties and Algorithms of the Hybrid Patch with B-spline Boundaries

Abstract

The hybrid patch proposed by Salvi effectively integrates both ribbon-based and control-based surfaces, thereby inheriting the advantages of each. This new multi-sided patch representation represents an improvement over Kato’s patch by incorporating interior control. It is also more versatile than the generalized Bézier surfaces, as it can handle positional and cross-derivative boundary constraints of arbitrary degrees. In this paper, we study the geometric properties and algorithms of the hybrid patch with B-spline boundaries. We first demonstrate that the hybrid patch possesses boundary interpolation and boundary derivative interpolation. The original degree elevation by Salvi changes the geometry of the patch slightly, which makes it undesirable in some applications. To address this issue, we propose an improved degree elevation algorithm for the hybrid patch, which preserves the geometric consistency of the patch. Furthermore, based on the knot insertion algorithm for B-splines, we propose a novel knot insertion algorithm for the hybrid patch with B-spline boundaries. Some representative examples show the effectiveness and validity of the proposed results.