A new method to construct polynomial minimal surfaces
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A new method to construct polynomial minimal surfaces Yong-Xia Hao1 Received: 2 July 2019 / Revised: 1 September 2020 / Accepted: 5 September 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract Minimal surface is an important type of surface with zero mean curvature. It exists widely in nature. The problem of finding all minimal surfaces presented in parametric form as polynomials is discussed by many authors. However, most of the constructions are based on the theorem that a harmonic surface with isothermal parameterization is minimal. As we all know, Weierstrass representation is a classical parameterization of minimal surfaces. Therefore, in this paper, we consider to construct polynomial minimal surfaces of arbitrary degree by Weierstrass representation. Moreover, there is a correspondence between our constructed polynomial minimal surfaces and Pythagorean hodograph curves. Several numerical examples are demonstrated to illustrate our results. Keywords Minimal surface · Weierstrass representation · Pythagorean hodograph curve Mathematics Subject Classification 65D17 · 65D18
1 Introduction Pythagorean hodograph (PH) curves, introduced by Farouki and Sakkalis (1990), are a special type of polynomial curves, which have the unique property that their parametric speed functions are also polynomials of the curve parameter (Kim and Moon 2017). The PH property enables us to compute the arc length of the curve exactly without numerical integration. Another important advantage of PH curves is that their offset curves are rational curves. So we do not need to rely on approximation algorithms for offset computation. These features make PH curves appropriate tools for many practical applications, such as computer-numericalcontrol machining and control of digital motion along curved paths (Wang and Fang 2009).
Communicated by Andreas Fischer. This work was supported by the National Natural Science Foundation of China (No. 11801225), University Science Research Project of Jiangsu Province (No. 18KJB110005) and the Research Foundation for Advanced Talents of Jiangsu University (No. 14JDG034).
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Yong-Xia Hao [email protected] Faculty of Science, Jiangsu University, Zhenjiang, China 0123456789().: V,-vol
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One may consult Farouki (2007) for more details on PH curves from algebraic frameworks to practical applications. Minimal surface is a field of study that has been studied extensively in many engineering and physics applications, such as object segmentation and mechanics of cellular materials (Hao and Li 2018; Monterde 2003, 2004; Nitsche 1989; Tråsdahl and Rønquist 2011; Xu et al. 2015a). It can be defined as the critical point of the area functional and, therefore, expressed by the variational equations or the equivalent Euler–Lagrange equations. The latter is also known as second-order elliptic equations. Weierstrass presented a general solution of the equation of minimal surfaces from another point of view without area concept. With Weier
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