A new class of curves of rational B-spline type

TL;DR

This paper introduces a novel class of rational B-spline functions based on a new rational parametrization, offering unique properties and potential for new curve designs, with a focus on their mathematical characteristics and asymptotic behavior.

Contribution

It presents a new family of rational B-splines dependent on a parameter, expanding the theoretical framework and properties of B-spline curves and functions.

Findings

The new rational B-splines satisfy positivity and partition of unity.

They form a true basis for approximating continuous functions.

Asymptotic behavior recovers classical regularity.

Abstract

A new class of rational parametrization has been developed and it was used to generate a new family of rational functions B-splines $\displaystyle{{\left({}^{\alpha}{\mathbf B}_{i}^{k} \right)}_{i=0}^{k}}$ which depends on an index $\alpha \in (-\infty,0)\cup (1,+\infty)$. This family of functions verifies, among other things, the properties of positivity, of partition of the unit and, for a given degree k, constitutes a true basis approximation of continuous functions. We loose, however, the regularity classical optimal linked to the multiplicity of nodes, which we recover in the asymptotic case, when $\alpha \longrightarrow \infty$. The associated \mbox{B-splines} curves verify the traditional properties particularly that of a convex hull and we see a certain "conjugated symmetry" related to $\alpha$. The case of open knot vectors without an inner node leads to a new family of rational Bezier curves that will be separately, object of in-depth analysis.