A class of αβγ-Bernstein–Bézier basis functions over triangular domain

doi:10.1016/j.amc.2013.06.043

Applied Mathematics and Computation

Volume 220, 1 September 2013, Pages 446-454

https://doi.org/10.1016/j.amc.2013.06.043 Get rights and content

Abstract

A class of $α β γ$ -Bernstein–Bézier basis functions over triangular domain, which include the cubic Ball basis functions over triangular domain and the cubic Bernstein–Bézier basis functions over triangular domain, is constructed. Based on these new basis functions, a kind of triangular Bernstein–Bézier-type patch with three exponential shape parameters is proposed. The shapes of the triangular Bernstein–Bézier-type patch can be modified intuitively and foreseeable by changing the values of the three exponential shape parameters under the same control net. The conditions for $G^{1}$ continuous smooth joining two triangular Bernstein–Bézier-type patches are given.

Introduction

It is well known that triangular Bernstein–Bézier patches, particular the triangular Bernstein–Bézier cubic patches, have attracted the attention of many researchers because of their great potentials of constructing complex shape, see [1], [2], [3], [4], [5]. However, the shapes of the triangular Bernstein–Bézier patches are fixed relatively to their control net. Although the weights in the rational Bernstein–Bézier basis over triangular domain, see [5], possess an effect for adjusting the shape of the patch, changing the weights to adjust the shape of a patch is quite opaque to the user.

In order to improve the shape of a patch and adjust the extent where a patch approaches its control net, some methods of generating triangular patch were presented by using shape parameters, see [6], [7], [8], [9] and the references quoted therein. In [6], the authors constructed a class of Bernstein–Bézier patch over triangular domain with a shape parameter. By changing the value of the shape parameter, different surfaces under the fixed control net can be obtained. In [7], Yang and Zeng proposed a class of triangular Bézier surfaces with $3 n (n + 1) / 2$ shape parameters. In [8], Yan and Liang constructed initial basis functions with a shape parameter over the triangular domain and then defined a basis functions of order n by a recursive approach. Based on the basis, they proposed a class of triangular Bernstein–Bézier-like surface with a shape parameter. In [9], by incorporating three exponential functions into the classical cubic Bernstein basis functions over triangular domain, a class of $λ μ η$ -Bernstein basis functions possessed three shape parameters over triangular domain was presented.

Recently, some researchers have put many efforts on the establishments of triangular Ball bases, see [10], [11], [12]. In [10], Goodman and Said defined bivariate Said-Ball basis on a triangle and designed triangular Said-Ball surface based on the research of Said-Ball curve. In [11], Hu et al. extended the univariate Wang-Ball basis to the bivariate cases on a triangle, designed triangular Wang-Ball surface, and pointed out its good value. In [12], Chen and Wang constructed a class of triangular DP surfaces.

The purpose of this paper is to present a class of $α β γ$ -Bernstein–Bézier basis functions over triangular domain, which include the cubic Ball basis functions over triangular domain and the cubic Bernstein-Bézier basis functions over triangular domain. The three exponential shape parameters in the Bernstein–Bézier-type patch serve as tension parameters and have a predictable adjusting role on the patch. The rest of this paper is organized as follows. Section 2 gives the construction and properties of the $α β γ$ -Bernstein–Bézier basis functions over triangular domain. In Section 3, the definition and properties of the triangular Bernstein–Bézier-type patch with three exponential shape parameters are presented. The conditions for $G^{1}$ continuous smooth joining two triangular Bernstein–Bézier-type patches are deduced. Conclusion is given in Section 4.

Section snippets

Construction of the $α β γ$ -Bernstein–Bézier basis functions

Firstly, we give the definition of the $α β γ$ -Bernstein–Bézier basis functions over triangular domain as follows.

Definition 1

Let $α, β, γ \in$ $[2, + \infty)$ , for $D = \{(u, v, w) |u + v + w = 1, u ⩾ 0, v ⩾ 0, w ⩾ 0\}$ , the following ten functions are defined as $α β γ$ -Bernstein–Bézier basis functions, with three exponential shape parameters $α, β$ and γ, over the triangular domain D: $\{\begin{matrix} B_{3, 0, 0}^{3} (u, v, w; α, β, γ) = u^{α}, \\ B_{0, 3, 0}^{3} (u, v, w; α, β, γ) = v^{β}, \\ B_{0, 0, 3}^{3} (u, v, w; α, β, γ) = w^{γ}, \\ B_{2, 1, 0}^{3} (u, v, w; α, β, γ) = u^{2} v [\frac{3 - 2 u - u^{α - 2}}{1 - u}], \\ B_{2, 0, 1}^{3} (u, v, w; α, β, γ) = u^{2} w [\frac{3 - 2 u - u^{α - 2}}{1 - u}], \\ B_{1, 2, 0}^{3} (u, v, w; α, β, γ) = v^{2} u [3 - \end{matrix}$

Definition and properties of the triangular Bernstein–Bézier-type patch

Definition 2

Let $α, β, γ \in$ $[2, + \infty)$ , given control points $P_{i, j, k} \in R^{3} (i, j, k \in N, i + j + k = 3)$ , and a triangular domain $D = \{(u, v, w) |u + v + w = 1, u ⩾ 0, v ⩾ 0, w ⩾ 0\}$ . We call $R (u, v, w) = \sum_{i + j + k = 3} B_{i, j, k}^{3} (u, v, w; α, β, γ) P_{i, j, k}, (u, v, w) \in D$ the triangular Bernstein-Bézier-type patch with three exponential shape parameters $α, β, γ$ .

From the properties of the $α β γ$ -Bernstein–Bézier basis functions over triangular domain, some properties of the triangular Bernstein–Bézier-type patch, analogous to that of the triangular Bernstein–Bézier cubic patch, can be

Conclusion

The $α β γ$ -Bernstein–Bézier basis functions over triangular domain constructed in this paper include the cubic Ball basis functions over triangular domain and the cubic Bernstein–Bézier basis functions over triangular domain as special cases. The three exponential shape parameters in the Bernstein–Bézier-type patch serve as tension parameters and have a predictable adjusting role on the patch. The conditions for $G^{1}$ continuous smooth joining two triangular Bernstein–Bézier-type patches are given.

Acknowledgements

We would like to thank the anonymous reviewers for their valuable remarks for improvements. The research is supported by the National Natural Science Foundation of China (No. 60970097, No. 11271376) and Graduate Students Scientific Research Innovation Project of Hunan Province (No. CX2012B111).

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A class of αβγ-Bernstein–Bézier basis functions over triangular domain

Abstract

Introduction

Section snippets

Construction of the αβγ-Bernstein–Bézier basis functions

Definition and properties of the triangular Bernstein–Bézier-type patch

Conclusion

Acknowledgements

J. Approx. Theory

Comput.-Aided Des.

Comput. Aided Geom. Des.

Comput. Aided Geom. Des.

Appl. Math. Comput.

Comput.-Aided Des.

Construction of the $α β γ$ -Bernstein–Bézier basis functions