Planar Typical Bézier Curves with a Single Curvature Extremum

Zhanchuan Cai,Shaolin Li,Gang Zhao,Aizeng Wang,Chuan He

doi:10.3390/math9172148

Zhanchuan Cai, Shaolin Li + Show 3 more

Open Access

https://doi.org/10.3390/math9172148

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Abstract

This paper focuses on planar typical Bézier curves with a single curvature extremum, which is a supplement of typical curves with monotonic curvature by Y. Mineur et al. We have proven that the typical curve has at most one curvature extremum and given a fast calculation formula of the parameter at the curvature extremum. This will allow designers to execute a subdivision at the curvature extremum to obtain two pieces of typical curves with monotonic curvature. In addition, we put forward a sufficient condition for typical curve solutions under arbitrary degrees for the G1 interpolation problem. Some numerical experiments are provided to demonstrate the effectiveness and efficiency of our approach.

Highlights

In CAGD (Computer Aided Geometric Design) applications, it is preferable to generate aesthetically pleasing surfaces, which are usually modeled by a set of feature curves with the required fairing shape
Considering the generality of objective curves for the interpolation problem, we relaxed the constraints of typical Bézier curves and provided a fast calculation of the parameter at the curvature extremum under arbitrary degrees
For the non-degeneration case, we aim to find out the relation between s and θ such that t∗ ∈ (0, 1), which means that κ (t) changes sign exactly once for t ∈ [0, 1] and the corresponding typical Bézier curve will have a single curvature extremum

Summary

Introduction

In CAGD (Computer Aided Geometric Design) applications, it is preferable to generate aesthetically pleasing surfaces, which are usually modeled by a set of feature curves with the required fairing shape. Wang et al gave other sufficient conditions for monotonic curvature of planar Bézier curves and B-spline curves without a transformation matrix [12,13]. Considering the generality of objective curves for the interpolation problem, we relaxed the constraints of typical Bézier curves and provided a fast calculation of the parameter at the curvature extremum under arbitrary degrees. Planar typical Bézier curves with Equation (3) have only one curvature extremum. Bib1i,k−2 decreases in [0, t∗] and increases in [t∗, 1] while bi b2i,k−3 keeps constant. Based on (I), (II) and (III), the curvature function of Equation (7) first increases and decreases, which indicates that κ(t) possesses a single curvature extremum for t ∈ [0, 1]. We show the subdivision at t∗ and prove that the two obtained segments belong to typical curves with monotonic curvature

Fast Parameter Formula

Subdivision at Curvature Extremum

Sufficient Condition for G1 Interpolation with Typical Bézier Curve Solution

Conclusions