Curve and surface construction based on the generalized toric-Bernstein basis functions

Jing-Gai Li,Chun-Gang Zhu

doi:10.1515/math-2020-0004

Jing-Gai Li, Chun-Gang Zhu

Open Access

https://doi.org/10.1515/math-2020-0004

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Abstract

Abstract The construction of parametric curve and surface plays an important role in computer aided geometric design (CAGD), computer aided design (CAD), and geometric modeling. In this paper, we define a new kind of blending functions associated with a real points set, called generalized toric-Bernstein (GT-Bernstein) basis functions. Then, the generalized toric-Bézier (GT-Bézier) curves and surfaces are constructed based on the GT-Bernstein basis functions, which are the projections of the (irrational) toric varieties in fact and the generalizations of the classical rational Bézier curves/surfaces and toric surface patches. Furthermore, we also study the properties of the presented curves and surfaces, including the limiting properties of weights and knots. Some representative examples verify the properties and results.

Highlights

In Computer Aided Geometric Design (CAGD) and Computed Aided Design (CAD), Bézier curves/surfaces play the central role [1, 2]
We de ne a new kind of blending functions associated with a real points set, called generalized toric-Bernstein (GT-Bernstein) basis functions
The generalized toric-Bézier (GT-Bézier) curves and surfaces are constructed based on the GT-Bernstein basis functions, which are the projections of the toric varieties and the generalizations of the classical rational Bézier curves/surfaces and toric surface patches

Summary

Introduction

In Computer Aided Geometric Design (CAGD) and Computed Aided Design (CAD), Bézier curves/surfaces play the central role [1, 2]. For given control points and weights, we can use the Bernstein basis functions to construct the classical rational Bézier curve.

Results

Conclusion