On the normalization of trigonometric and hyperbolic B-splines

2025/08/03 by Hendrik Speleers, Speleers, Hendrik
Engineering · Computer Science · Mathematics · #Advanced Numerical Analysis Techniques #Polynomial and algebraic computation #Mathematical functions and polynomials

paper · doi:10.48550/arxiv.2508.01817

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≥3, with a variable number of control points. We also illustrate the approximation power of trigonometric and hyperbolic splines.

Citations

• A general class of C¹ smooth rational splines: Application to construction of exact ellipses and ellipsoids
• Critical length: an alternative approach

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