Multi-sided Surfaces with Fullness Control

Multi-sided surfaces are important in several areas of Computer-Aided Geometric Design, including curve network based design, approximation of triangular meshes, hole filling, and so on. In the majority of surface representations, the boundary constraints entirely determine the interior of these patches, however, frequently there is a need to have additional design freedom for shaping the surface. In this paper, we investigate new multi-sided surfaces, where such interior control is possible. As a reference, we take a recent variant of the Gregory patch,8 and compare it with two new representations: (i) a transfinite patch defined by boundary ribbons (curves and cross-derivatives) and a single surface point to set in the middle, and (ii) a recently published, new n-sided Bézier patch,10 defined by a grid of control points. We will discuss the different degrees of freedom for shape adjustment, and illustrate our results by a few examples showing mesh approximation quality.