On the normalization of trigonometric and hyperbolic B-splines

TL;DR

This paper develops explicit normalization methods for trigonometric and hyperbolic B-splines, enabling their use in design and approximation tasks by ensuring they form a partition of unity.

Contribution

It introduces a recursive algorithm for normalization weights and demonstrates applications in exact circle representation and spline approximation.

Findings

Normalized B-splines form a partition of unity

Exact circle representation with trigonometric B-splines

Effective approximation capabilities of these splines

Abstract

Trigonometric and hyperbolic B-splines can be computed via recurrence relations analogous to the classical polynomial B-splines. However, in their original formulation, these two types of B-splines do not form a partition of unity and consequently do not admit the notion of control polygons with the convex hull property for design purposes. In this paper, we look into explicit expressions for their normalization and provide a recursive algorithm to compute the corresponding normalization weights. As example application, we consider the exact representation of a circle in terms of $C^{2n-1}$ trigonometric B-splines of order $m=2n+1\geq3$, with a variable number of control points. We also illustrate the approximation power of trigonometric and hyperbolic splines.