An Application of Variational Minimization: Quasi-Harmonic Coons Patches

Abstract

For a minimal surface, the mean curvature of the surface vanishes for all possible parameterizations which results in a second-order nonlinear partial differential equation ( p d e ), whose solution in general is the desired surface as the unknown function of surface parameters. The solution of this partial differential equation is known only for very few cases. Instead of solving the corresponding partial differential equation, we exploit an ansatz method (Ahmad et al. (2013), Ahmad et al. (2014) Ahmad et al. (2015)), used for Coons patches spanned by finite number of boundary curves for a quasi-minimal surface as the extremal of r m s of mean curvature. The ansatz method targets a slightly perturbed surface (the rational blending interpolants-based Coons patch, Hermite cubic polynomials interpolants-based Coons patch, and Ferguson surface (vanishing twist vectors) in our case) that comprises initially a nonminimal surface plus the product of a real parameter with a variational function of surface parameters (vanishes at the boundary curves along the unit normal to the slightly perturbed surface). The variational function can be deliberately chosen as the product of linear functions and the mean curvature of the initial nonminimal surface such that it is zero at the boundary curves and then replace this mean curvature by the mean curvature of the resulting surface to find a surface of reduced area, and the process can be repeated for further improvement. In this article, we extend the ansatz method (1) for the rational blending functions interpolants-based Coons patch with a parameter in their form for its different values and (2) for the blending functions comprising Hermite cubic polynomial interpolants-based Coons patch (bicubically blended Coons patch (BBCP) and the Ferguson surface). The ansatz method can be extended for the variational extremal of the surfaces for fuzzy optimal control problems (Filev et al. (1992), Emamizadeh (2005), Farhadinia (2011), Mustafa et al. (2021)).

Geometric analysis of non-degenerate shifted-knots Bézier surfaces in Minkowski space

“…Despite decades of research on Be ´zier curves and surfaces, there is still much to learn about them, and their use continues to be essential in the representation and communication of geometric data. In recent years, Be ´zier curves and surfaces in the Euclidean space-E 3 have been extensively studied, as demonstrated by various publications such as [1][2][3][4][5][6][7][8][9][10][11][12]. These studies have focused on different aspects of Be ´zier curves and surfaces, including their properties, applications, and computational methods.…”

Extremal Solutions for Surface Energy Minimization: Bicubically Blended Coons Patches

“…These methods are elaborated upon in the subsequent section for reference. In our previous work [48], we have explained the method of finding the twist vectors for the BBCP in Section 5; specifically, Adini's method is given in Equations ( 50) and (51) for the twist vectors, utilizing Equation (49) for the tangent vectors. The respective twist vectors are given in Equation (52) of the same reference [48].…”

“…In our previous work [48], we have explained the method of finding the twist vectors for the BBCP in Section 5; specifically, Adini's method is given in Equations ( 50) and (51) for the twist vectors, utilizing Equation (49) for the tangent vectors. The respective twist vectors are given in Equation (52) of the same reference [48]. For a more comprehensive understanding of the twist vectors, we recommend referring to section (16.3) of ref.…”

A Computational Model for q‐Bernstein Quasi‐Minimal Bézier Surface

“…e vanishing condition of the gradient of the functional gives linear algebraic constraints on the unknown control points in terms of known boundary control points. One of the widely used class of surfaces is known as the Bézier surfaces based on Bernstein polynomials as has been done by Monterde [19,20], and he found the quasi-minimal Bézier surface as the extremal of Dirichlet functional, other related works for ansatz method, and for the vanishing condition of certain energy functionals for the Bézier surfaces based on modified Bernstein polynomials and Coons patch one can see Ahmad et al [33][34][35][36][37][38].…”