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

## TL;DR

This paper introduces generalized toric-Bernstein basis functions to construct new parametric curves and surfaces, extending classical Bézier methods with properties rooted in toric geometry, useful in geometric design.

## Contribution

It defines a new class of blending functions called GT-Bernstein basis functions and constructs associated curves and surfaces, generalizing classical rational Bézier forms with toric variety insights.

## Key findings

- The GT-Bernstein basis functions generalize classical Bézier basis functions.
- Constructed curves and surfaces exhibit desirable geometric properties.
- Examples confirm the theoretical properties and potential applications.

## Abstract

The construction of parametric curve and surface plays 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-Bezier (GT-B\'ezier) 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 B\'ezier curves and 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.