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The classification of bi-quintic parametric polynomial minimal surfaces

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Indexed by:期刊论文

Date of Publication:2017-03-01

Journal:APPLIED MATHEMATICS-A JOURNAL OF CHINESE UNIVERSITIES SERIES B

Included Journals:SCIE、Scopus

Volume:32

Issue:1

Page Number:14-26

ISSN No.:1005-1031

Abstract:Parametric polynomial surface is a fundamental element in CAD systems. Since the most of the classic minimal surfaces are represented by non-parametric polynomial, it is interesting to study the minimal surfaces represented in parametric polynomial form. Recently, Ganchev presented the canonical principal parameters for minimal surfaces. The normal curvature of a minimal surface expressed in these parameters determines completely the surface up to a position in the space. Based on this result, in this paper, we study the bi-quintic isothermal minimal surfaces. According to the condition that any minimal isothermal surface is harmonic, we can acquire the relationship of some control points must satisfy. Follow up, we obtain two holomorphic functions f(z) and g(z) which give the Weierstrass representation of the minimal surface. Under the constrains that the minimal surface is bi-quintic, f(z) and g(z) can be divided into two cases. One case is that f(z) is a constant and g(z) is a quadratic polynomial, and another case is that the degree of f(z) and g(z) are 2 and 1 respectively. For these two cases, we transfer the isothermal parameter to canonical principal parameter, and then compute their normal curvatures and analyze the properties of the corresponding minimal surfaces. Moreover, we study some geometric properties of the bi-quintic harmonic surfaces based on the Bezier representation. Finally, some numerical examples are demonstrated to verify our results.

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李彩云，女，1984年生，大连理工大学数学科学学院副教授，硕士生导师，主要从事计算几何与计算机辅助设计方向的研究工作，内容涉多元样条、曲线曲面造型的理论和应用研究等。到目前为止，在国内外重要期刊发表论文20余篇,主持1项国家自然科学基金青年基金项目，参加了包括2项国家自然科学基金面上项目在内的多个科研与教学项目建设。2017年入选大连理工大学第四届“星海骨干”人才培育计划。近年发表部分论文如下：
[1] 朱春钢, 李彩云, 王仁宏，异度隐函数样条曲线曲面, 计算机辅助设计与图形学学报, 2009, 21(7), 930-935.
[2] C.Y. Li, C.G. Zhu, A multilevel univariate cubic spline quasi-interpolation and application to numerical integration, Mathematical Methods in the Applied Sciences, 2010, 33(13), 1578-1586.
[3] 李彩云, 朱春钢, 王仁宏, 参数曲线的分段近似隐式化, 高校应用数学学报, 2010, 25(2), 202-210.
[4] C.Y. Li, R.H. Wang, C.G. Zhu, Designing and G^1 connection of developable surfaces through Bézier geodesics, Applied Mathematics and Computation, 2011, 218(7), 3199-3208.
[5] C.Y. Li, R.H. Wang, C.G. Zhu, Parametric representation of a surface pencil with a common spatial line of curvature, Computer-Aided Design, 2011, 43(9) , 1110-1117.
[6] H.Y. Liu, C.G. Zhu, C.Y. Li, Constructing N-sided toric surface patches from boundary curves, Journal of Information and Computational Science, March, 2012, 9(3), 737-743.
[7] C.Y. Li, R.H. Wang, C.G. Zhu, Designing approximation minimal surfaces with geodesics, Applied Mathematical Modelling, 2013, 37 (9), 6415-6424.
[8] C.Y. Li, R.H. Wang, C.G. Zhu, An approach for designing a developable surface through a given line of curvature, Computer-Aided Design, 2013, 45 (3) , 621-627.
[9] C.Y. Li, R.H. Wang, C.G. Zhu, A generalization of surface family with common line of curvature, Applied Mathematics and Computation, May, 2013, 219 (17), 9500-9507.
[10] Cai-Yun Li; Chun-Gang Zhu*; Ren-Hong Wang, Spacelike developable surfaces through a common line of curvature in Minkowski 3-space, Journal of Advanced Mechanical Design, Systems, and Manufacturing, 2015, 9(4), JAMDSM0050.
[11] 李彩云, 朱春钢, 王仁宏, 插值特殊曲线的曲面造型研究, 中国科学：数学，“庆贺徐利治教授95华诞专辑”，2015，45(9)，1441-1456.
[12] 李彩云, 项昕，朱春钢，一种插值曲率线的直纹面可展设计方法, 中国图像图形学报, 2016, 21(4), 527-531。
[13] 王慧，朱春钢，李彩云, 六次PH曲线G2 Hermite插值, 图学学报, 2016, 37 (2), 155-165.
[14] LI Cai-yun, ZHU Chun-gang, The classification of bi-quintic parametric polynomial minimal surfaces, Appl. Math. J. Chinese Univ, 2017, 32(1), 14-26.
[15] Cai-Yun Li*, Chun-Gang Zhu, G1 continuity of four pieces of developable surfaces with Bezier boundaries, Journal of Computational and Applied Mathematics, 2018, 329, 280-293.
[16] Hui Wang, Chungang Zhu*, Caiyun Li, Identication and Hermite interpolation of planar sextic Pythagorean-hodograph curves, Journal of Mathematical Research with Applications. 2017, 37(1), 59-72.

[17] 王慧，朱春钢，李彩云，插值有理Bezier渐近四边形的有理Bezier曲面，计算机辅助几何设计与图与形学学报，2017, 29(8), 1497-1504.

[18] Hui Wang, Chun-Gang Zhu*, Cai-Yun Li, The design of Bezeir surface through quintic Bezier asymptotic quadrilateral, Journal of Computational Mathematics, 37(5)(2019),723-740.

[19] Cai-Yun Li, Chungang Zhu*, Designing Developable C-Bézier Surface with Shape Parameters, Mathematics, 8, 402 (2020), doi:10.3390/math8030402.

[20] Caiyun Li, Chungang Zhu*, Construction of the spacelike constant angle surface family in Minkowski 3-space, AIMS Mathematics, 2020,5(6): 6341-6354.

[21] Wei Meng,Caiyun Li*, Qianqian Liu, Geometric Modeling of C-Bezier curve and surface with shape parameters, Mathematics, 2021, 9, 2651, https://doi.org/10.3390/math9212651.