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On the construction of polynomial minimal surfaces with Pythagorean normals
Applied Mathematics and Computation ( IF 4 ) Pub Date : 2022-08-08 , DOI: 10.1016/j.amc.2022.127439
Rida T. Farouki, Marjeta Knez, Vito Vitrih, Emil Žagar

A novel approach to constructing polynomial minimal surfaces (surfaces of zero mean curvature) with isothermal parameterization from Pythagorean triples of complex polynomials is presented, and it is shown that they are Pythagorean normal (PN) surfaces, i.e., their unit normal vectors have a rational dependence on the surface parameters. This construction generalizes a prior approach based on Pythagorean triples of real polynomials, and yields more free shape parameters for surfaces of a specified degree. Moreover, when one of the complex polynomials is just a constant, the minimal surfaces have the Pythagorean–hodograph (PH) preserving property — a planar PH curve in the parameter domain is mapped to a spatial PH curve on the surface. Cubic, quartic and quintic examples of these minimal PN surfaces are presented, including examples of solutions to the Plateau problem, with boundaries generated by planar PH curve segments in the parameter domain. The construction is also generalized to the case of minimal surfaces with non–isothermal parameterizations. Finally, an application to the problem of interpolating three given points in R3 as the corners of a triangular cubic minimal surface patch, such that the three patch sides have prescribed lengths, is addressed.



中文翻译:

用毕达哥拉斯法线构造多项式最小曲面

提出了一种从复多项式的毕达哥拉斯三元组等温参数化构造多项式最小曲面(平均曲率为零的曲面)的新方法,并证明它们是毕达哥拉斯法线(PN)曲面,即它们的单位法向量具有有理依赖于表面参数。这种构造概括了基于实数多项式的毕达哥拉斯三元组的先前方法,并为指定度数的表面产生更多自由形状参数。此外,当一个复多项式只是一个常数时,最小曲面具有毕达哥拉斯曲线 (PH) 保持特性——参数域中的平面 PH 曲线映射到曲面上的空间 PH 曲线。介绍了这些最小 PN 曲面的三次、四次和五次示例,包括解决高原问题的示例,边界由参数域中的平面 PH 曲线段生成。该构造也推广到具有非等温参数化的最小表面的情况。最后,一个应用程序对三个给定点进行插值R3作为三角形立方最小表面补丁的角,使得三个补丁边具有规定的长度,被解决。

更新日期:2022-08-09
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