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On the normalization of trigonometric and hyperbolic B-splines

Hendrik Speleers

TL;DR

The paper tackles the limitation that trig and hyperbolic B-splines do not form a partition of unity in their natural form, hindering design and analysis. It introduces new explicit normalization weights and a recursive algorithm to compute them, with efficient simplifications for uniform knots via $q$-binomial coefficients. The approach is demonstrated through exact circle representations using $C^{2n-1}$ trig B-splines and by evaluating the approximation power of trig and hyperbolic splines, highlighting improved accuracy per degree of freedom for high-order splines. The results enhance the practical usability of these splines in design and isogeometric analysis and suggest potential integration into MDTB-splines tooling.

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.

On the normalization of trigonometric and hyperbolic B-splines

TL;DR

The paper tackles the limitation that trig and hyperbolic B-splines do not form a partition of unity in their natural form, hindering design and analysis. It introduces new explicit normalization weights and a recursive algorithm to compute them, with efficient simplifications for uniform knots via -binomial coefficients. The approach is demonstrated through exact circle representations using trig B-splines and by evaluating the approximation power of trig and hyperbolic splines, highlighting improved accuracy per degree of freedom for high-order splines. The results enhance the practical usability of these splines in design and isogeometric analysis and suggest potential integration into MDTB-splines tooling.

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 trigonometric B-splines of order , with a variable number of control points. We also illustrate the approximation power of trigonometric and hyperbolic splines.

Paper Structure

This paper contains 6 sections, 2 theorems, 74 equations, 8 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Definition of normalized B-splines
  3. Computation of the normalization weights
  4. Normalized B-spline examples
  5. Concluding remarks
  6. Proof of the identity \ref{['eq:B-spline-trig-weights-int']}.

Key Result

Theorem 3.1

Let $m=2n+1\geq3$. The normalization weight in eq:B-spline-trig-weights-full can be expressed as and similarly, the normalization weight in eq:B-spline-hypb-weights-full can be expressed as

Figures (8)

  • Figure 1: Cardinality of the sets $\mathcal{Q}_n$, $\widehat{\mathcal{Q}}_n$, $\mathcal{S}_n$ for different values of $n$ (in logarithmic scale).
  • Figure 2: Visualization of sets of normalized trigonometric B-splines ${N}^T_{j,m}$ defined on the knot sequence \ref{['eq:ex-B-splines-knots']} for different values of $m$ as described in Example \ref{['ex:B-splines']}. The use of open knots leads to interpolation at the end points of the interval.
  • Figure 3: Visualization of sets of normalized hyperbolic B-splines ${N}^H_{j,m}$ defined on the knot sequence \ref{['eq:ex-B-splines-knots']} for different values of $m$ as described in Example \ref{['ex:B-splines']}. The use of open knots leads to interpolation at the end points of the interval.
  • Figure 4: Representation of a full circle by means of a normalized trigonometric B-spline curve of order $m=3$ defined on the knot sequence \ref{['eq:ex-circle-knots']} for $p=4,8$ as described in Example \ref{['ex:circle']} ($\theta=\pi/p$). The corresponding control points are indicated with the symbol $\star$ and form a regular $p$-sided polygon that is tangential to the circle.
  • Figure 5: Representation of a full circle by means of a normalized trigonometric B-spline curve of order $m=5,7$ defined on the knot sequence \ref{['eq:ex-circle-knots-m']} for $p=8$ as described in Example \ref{['ex:circle-m']} ($\theta=\pi/p$). The corresponding control points are indicated with the symbol $\star$ and form a regular $p$-sided polygon.
  • ...and 3 more figures

Theorems & Definitions (17)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Theorem 3.1
  • proof
  • Example 3.1
  • Example 3.2
  • Remark 3.1
  • Remark 3.2
  • Theorem 3.2
  • ...and 7 more