A class of triangular surface of the same degree with four shape parameters

doi:10.3785/j.issn.1008-9497.2023.02.004

Journal of Zhejiang University (Science Edition)

2023, Vol. 50

Issue (2): 153-159 DOI: 10.3785/j.issn.1008-9497.2023.02.004

Mathematics and Computer Science

A class of triangular surface of the same degree with four shape parameters

Xiang KONG,Jun CHEN(

)

Faculty of Science，Ningbo University of Technology，Ningbo 315211，Zhejiang Province，China

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Abstract

4 independent new shape parameters were incorporated into the binary polynomial basis functions over triangular domain without raising the degree, therefore introducing a new class of triangular surface with the same degree. The shape of the new triangular surface can be modified without adjusting the positions of the control points, meantime preserving the excellent geometric properties similar to the triangular Bézier surface. Moreover, the triangular Bézier surface and some triangular surfaces with shape parameters from previous studies can be regarded as the special cases of our work. With the aid of the shape parameters, the shape of the triangular surface could be adjusted even if its boundary curves keep unchanged. Finally, the conditions for G1 continuous smooth joining between two adjacent triangular patches are derived.

Key words： triangular surface shape parameter triangular domain surface design geometric continuity

Received: 14 February 2022 Published: 21 March 2023

CLC:

TP 391

Corresponding Authors: Jun CHEN E-mail: chenj@nbut.edu.cn

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Cite this article:

Xiang KONG,Jun CHEN. A class of triangular surface of the same degree with four shape parameters. Journal of Zhejiang University (Science Edition), 2023, 50(2): 153-159.

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https://www.zjujournals.com/sci/EN/Y2023/V50/I2/153

一类带4个形状参数的同次三角曲面构造算法

为构造三角域上二元多项式基函数，在不提高多项式次数的前提下，添加4个独立的形状参数，由此得到一类新的同次三角曲面。这类三角曲面既保留了三角Bézier曲面的几何特性，又能在保持控制顶点不变时进行微调。已有的三角Bézier曲面以及部分带形状参数的三角曲面均为本文的特例。4个独立的形状参数对曲面外形有不同的影响，其中2个形状参数在3条边界固定时仍能修改三角曲面外形。最后，讨论了2张三角曲面在进行G¹连续拼接时形状参数所需满足的条件。

关键词： 三角曲面, 形状参数, 三角域, 曲面设计, 几何连续

Fig.1 10 control points of the triangular surface

曲面来源	独立形状参数的数量	本文算法特例
三角Bézier曲面^［1］	0	$α 1 = α 4 = 1, α 2 = α 3 = 0$
文献［11］	1	$α 1 = 2 + α 3, α 2 = α 3 = 0, α 4 = 1$
文献［12］	2	$α 1 = λ 1 I I 3, α 2 = α 3 = 0, α 4 = λ 2 I I 2$
文献［13］	1	$α 1 = 2 + λ 3, α 2 = α 3 = 0, α 4 = 1$
文献［14］	1	$α 1 = α 4 = λ, α 2 = 0, α 3 = 1 - λ 6$
文献［15］	1	$α 1 = 2 + λ 3, α 2 = 1 - λ 3, α 3 = 0, α 4 = 1$

Table 1 Comparison with the past results when n = 3

Fig. 2 The trend of the points when α₃changes

Fig. 3 Triangular surfaces S （u，v，w） when α₂=α₃=0， α₄=1

Fig.4 Triangular surfaces S （u，v，w） when α₁=0.6， α₃=0， α₄=1

Fig.5 Triangular surfaces S （u，v，w） when α₁=1， α₂=0， α₄=0.2

Fig.6 Triangular surfaces S （u，v，w） when α₁=0.5， α₂=α₃=0

Fig. 7 The control points of two adjacent triangular patches

Fig.8 G¹ continuity of two adjacent patches


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