On multi-degree splines

TL;DR

This paper introduces new methods for constructing and evaluating multi-degree splines, which are piecewise polynomial functions with sections of varying degrees, enhancing their applicability in geometric modeling.

Contribution

It extends existing approaches by proposing a novel transition function-based method for B-spline basis construction and evaluation, enabling more flexible multi-degree spline design.

Findings

Developed integral recurrence relations for B-spline basis construction.

Proposed an alternative transition function-based method for spline evaluation.

Demonstrated algorithms for knot-insertion, degree elevation, and Bézier conversion.

Abstract

Multi-degree splines are piecewise polynomial functions having sections of different degrees. For these splines, we discuss the construction of a B-spline basis by means of integral recurrence relations, extending the class of multi-degree splines that can be derived by existing approaches. We then propose a new alternative method for constructing and evaluating the B-spline basis, based on the use of so-called transition functions. Using the transition functions we develop general algorithms for knot-insertion, degree elevation and conversion to B\'ezier form, essential tools for applications in geometric modeling. We present numerical examples and briefly discuss how the same idea can be used in order to construct geometrically continuous multi-degree splines.