Building geometrically continuous splines

TL;DR

This paper introduces a modern framework for modeling polygonal surfaces with geometric continuity, providing definitions, basis structures, dimension formulas, and practical insights for spline functions with G1 gluing data.

Contribution

It offers a comprehensive, practical approach to geometric continuity in spline modeling, including new definitions, basis structures, and a dimension formula for bounded degree spline spaces.

Findings

Defined general structure for spline basis

Proved dimension formula for spline spaces

Presented a comprehensive example

Abstract

With the renewed and growing interest in geometric continuity in mind, this article gives a general definition of geometrically continuous polygonal surfaces and geometrically continuous spline functions on them. Polynomial splines defined by G1 gluing data in terms of rational functions are analyzed further. A general structure for a spline basis is defined, and a dimension formula is proved for spline spaces of bounded degree on polygonal surfaces made up of rectangles and triangles. Lastly, a comprehensive example is presented, and practical perspectives of geometric continuity are discussed. The whole objective of the paper is to put forward a modernized, practicable framework of modeling with geometric continuity.