A multisided [formula omitted] B-spline patch over extraordinary vertices in quadrilateral meshes

Gerben J Hettinga,Jiří Kosinka

doi:10.1016/j.cad.2020.102855

Abstract

We propose a generalised B-spline construction that extends uniform bicubic B-splines to multisided regions spanned over extraordinary vertices in quadrilateral meshes. We show how the structure of the generalised Bézier patch introduced by Várady et al. can be adjusted to work with B-spline basis functions. We create ribbon surfaces based on B-splines using special basis functions. The resulting multisided surfaces are C2 continuous internally and connect with G2 continuity to adjacent regular and other multisided B-splines patches. We visually assess the quality of these surfaces and compare them to Catmull–Clark limit surfaces on several challenging geometrical configurations.

Highlights

Constructing arbitrary topology surfaces based on B-splines has been a long-standing challenge in geometric design and computer graphics, and recently in analysis
Many of the commonly used subdivision surfaces generalise B-spline surfaces governed by quad-dominant meshes by creating a smooth surface by an iterative mesh refinement process
For the multisided B-spline construction we look for a generalisation of uniform cubic B-splines to multisided regions, in our case to the one-ring neighbourhoods of extraordinary vertices

Summary

Introduction

Constructing arbitrary topology surfaces based on B-splines has been a long-standing challenge in geometric design and computer graphics, and recently in (isogeometric) analysis. Extraordinary vertices and faces introduce problems in the definition of the basis functions which are typically only well defined over regular control mesh regions. Many of the commonly used subdivision surfaces generalise B-spline surfaces governed by quad-dominant meshes by creating a smooth surface by an iterative mesh refinement process. This process extends standard uniform B-spline subdivision masks/stencils to faces with any number of sides (extraordinary faces) and vertices with valency other than four (extraordinary vertices). Subdivision surfaces are typically only G1 continuous at extraordinary vertices and introduce other undesirable artefacts in the surrounding area, especially over high valency regions. A simple technique that preserves both good shape and offers continuity higher than G1 has not been found yet [2]

Results

Discussion

Conclusion