An approximation method based on MRA for the quasi-Plateau problem

Hao, Yong-Xia; Li, Chong–Jun; Wang, Ren-Hong

doi:10.1007/s10543-012-0412-2

Cited by 5 publications

(5 citation statements)

References 25 publications

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“…The surface given by (24) for 0 ≤ u, v ≤ 1 is a non-minimal surface spanned by a boundary composed of non-coplanar straight lines. The fundamental magnitudes of this initial surface are For n = 0, eq.…”

Section: The Bilinear Interpolant (24) Spanned By Four Boundary Nomentioning

confidence: 99%

“…Xu et al [23] study approximate developable surfaces and approximate minimal surfaces (defined as the minimum of the norm of mean curvature) and obtain tensor product Bézier surfaces using a nonlinear optimization algorithm. Hao et al [24] find the parametric surface of minimal area defined on a rectangular parameter domain among all the surfaces with prescribed borders using an approximation based on Multi Resolution Method using B-splines. Xu and Wang [25] study quintic parametric polynomial minimal surfaces and their properties.…”

Section: Introductionmentioning

confidence: 99%

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Near-stability of a quasi-minimal surface indicated through a tested curvature algorithm

Ahmad

Masud

2015

Computers & Mathematics with Applications

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We decrease the rms mean curvature and area of a variable surface with a fixed boundary by iterating a few times through a curvature-based variational algorithm. For a boundary with a known minimal surface, starting with a deliberately chosen non-minimal surface, we achieve up to 65 percent of the total possible decrease in area. When we apply our algorithm to a bilinear interpolant bounded by four non-coplanar straight lines, the area decrease by the same algorithm is only 0.116179 percent of the original value. This relative stability suggests that the bilinear interpolant is already a quasi-minimal surface.

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Section: The Bilinear Interpolant (24) Spanned By Four Boundary Nomentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Near-stability of a quasi-minimal surface indicated through a tested curvature algorithm

Ahmad

Masud

2015

Computers & Mathematics with Applications

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“…Furthermore, Be ´zier surfaces have been employed to solve the Plateau-Be ´zier problem, which involves finding a Be ´zier surface of minimal area among all possible surfaces spanned by the same boundary [2,18,19]. The utilization of variational techniques, including the extremization of Dirichlet and energy functionals, to tackle the Plateau-Be ´zier problem has been significant [7,[20][21][22][23]. Additionally, numerical methods and energy functionals have been applied to obtain quasi-minimal surfaces [3, 5-7, 9, 24-27].…”

Section: Introductionmentioning

confidence: 99%

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

Bashir,

Ahmad

2024

PLoS ONE

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In this paper, we investigate the properties of timelike and spacelike shifted-knots Bézier surfaces in Minkowski space-E 1 3. These surfaces are commonly used in mathematical models for surface formation in computer science for computer-aided geometric design and computer graphics, as well as in other fields of mathematics. Our objective is to analyze the characteristics of timelike and spacelike shifted-knots Bézier surfaces in Minkowski space-E 1 3. To achieve this, we compute the fundamental coefficients of shifted-knots Bézier surfaces, including the Gauss-curvature, mean-curvature, and shape-operator of the surface. Furthermore, we present numerical examples of timelike and spacelike bi-quadratic (m = n = 2) and bi-cubic (m = n = 3) shifted-knots Bézier surfaces in Minkowski space-E 1 3 to demonstrate the applicability of the technique in Minkowski space.

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“…The Plateau-Bézier problem for triangular Bézier patches was studied by Arnal et al [27]. Hao et al [28,29] dealt with the quasi-Plateau-Bézier problem for a broader variety of boundaries consisting of polynomial boundary curves obtaining Bézier surfaces as the solution of the extremal of Dirichlet functional, the harmonic and biharmonic functionals, and Multiresolution Analysis (MRA) employing B-splines. The properties related to the minimal surfaces spanned by a boundary with quintic form of parametric polynomials can be seen in Xu and Wang's study [30].…”

Section: Introductionmentioning

confidence: 99%

Variationally Improved Bézier Surfaces with Shifted Knots

Ahmad

Hassan

Mahmood

et al. 2021

Advances in Mathematical Physics

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The Plateau-Bézier problem with shifted knots is to find the surface of minimal area amongst all the Bézier surfaces with shifted knots spanned by the admitted boundary. Instead of variational minimization of usual area functional, the quasi-minimal Bézier surface with shifted knots is obtained as the solution of variational minimization of Dirichlet functional that turns up as the sum of two integrals and the vanishing condition gives us the system of linear algebraic constraints on the control points. The coefficients of these control points bear symmetry for the pair of summation indices as well as for the pair of free indices. These linear constraints are then solved for unknown interior control points in terms of given boundary control points to get quasi-minimal Bézier surface with shifted knots. The functional gradient of the surface gives possible candidate functions as the minimizers of the aforementioned Dirichlet functional; when solved for unknown interior control points, it results in a surface of minimal area called quasi-minimal Bézier surface. In particular, it is implemented on a biquadratic Bézier surface by expressing the unknown control point P 11 as the linear combination of the known control points in this case. This can be implemented to Bézier surfaces with shifted knots of higher degree, as well if desired.

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An approximation method based on MRA for the quasi-Plateau problem

Cited by 5 publications

References 25 publications

Near-stability of a quasi-minimal surface indicated through a tested curvature algorithm

Near-stability of a quasi-minimal surface indicated through a tested curvature algorithm

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

Variationally Improved Bézier Surfaces with Shifted Knots

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