C1 Smooth Triangular Surface Patch Constructed by C-curves

Li, Wei; Hagiwara, Ichiro; Wu, Zhuoqi

doi:10.1299/jsmec.48.159

JSME Int. J., Ser. C

2005

DOI: 10.1299/jsmec.48.159

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C1 Smooth Triangular Surface Patch Constructed by C-curves

Wei Li

Ichiro Hagiwara

Zhuoqi Wu

Abstract: In this paper, a method of constructing C 1 triangular patch using C-curves is presented. C-curves are developed by using the basis {sint,cost,t,1} by Zhang(1) -(3) , Pottmann and Wagner (4) , and it overcomes some shortcomings of the non-uniform rational B-splines (NURBS) model. For example, it can represent exactly transcendental curves such as the helix and the cycloid. So, how to develop C-curve into surface becomes a very important problem. In Ref. (3), tensor product quadrilateral patches have been const… Show more

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“…In 2004, Chen and Wang [4] extended them to the general algebraic trigonometric polynomial space. Meanwhile, the properties and applications of the C-Bézier were widely studied in many documents [6][7][8]11,13]. There exists similar theory for both the algebraic trigonometric space and the algebraic hyperbolic space.…”

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Triangular domain extension of algebraic trigonometric Bézier-like basis

Wei

Shen

Wang

2011

Appl. Math. J. Chin. Univ.

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In computer aided geometric design (CAGD), Bézier-like bases receive more and more considerations as new modeling tools in recent years. But those existing Bézier-like bases are all defined over the rectangular domain. In this paper, we extend the algebraic trigonometric Bézier-like basis of order 4 to the triangular domain. The new basis functions defined over the triangular domain are proved to fulfill non-negativity, partition of unity, symmetry, boundary representation, linear independence and so on. We also prove some properties of the corresponding Bézier-like surfaces. Finally, some applications of the proposed basis are shown. §1 IntroductionIn computer aided geometric design (CAGD), the classical Bernstein-Bézier system defined for polynomial space has become an important tool for free form curve/surface modeling since 1960s. In recent years, the Bézier-like basis defined for the blended spaces has received attention. The most important reason is that Bézier-like bases can exactly represent many analytic curves and surfaces. However the NURBS representations have complicated rational forms. In 1990s, Zhang [14,15] presented the Bézier-like curves, called C-Bézier, for the space Γ =span{1, t, sin t, cos t}. In 2004, Chen and Wang [4] extended them to the general algebraic trigonometric polynomial space. Meanwhile, the properties and applications of the C-Bézier were widely studied in many documents [6][7][8]11,13]. There exists similar theory for both the algebraic trigonometric space and the algebraic hyperbolic space. And many scholars, for example, Fang, Xu and Zhang [5,12,16], unified these two spaces by introducing new parameters, initial functions and complex numbers.The above mentioned papers on Bézier-like curves and surface are all confined in the rectangular domain. However, curves and surfaces modeling over the triangular domain are also important, just like Bernstein-Bézier system [1-3]. Furthermore, Shen and Wang [9,10] have

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Triangular domain extension of algebraic trigonometric Bézier-like basis

Wei

Shen

Wang

2011

Appl. Math. J. Chin. Univ.

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Triangular domain extension of linear Bernstein-like trigonometric polynomial basis

Shen

Wang

2010

J. Zhejiang Univ. - Sci. C

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C1 Smooth Triangular Surface Patch Constructed by C-curves

Cited by 2 publications

References 13 publications

Triangular domain extension of algebraic trigonometric Bézier-like basis

Triangular domain extension of algebraic trigonometric Bézier-like basis

Triangular domain extension of linear Bernstein-like trigonometric polynomial basis

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