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24 pages, 6364 KB

Open AccessArticle

Bezier Curves and Surfaces with the Generalized α-Bernstein Operator

by Davut Canlı and Süleyman Şenyurt

Symmetry 2025, 17(2), 187; https://doi.org/10.3390/sym17020187 - 25 Jan 2025

Cited by 1 | Viewed by 776

In the field of Computer-Aided Geometric Design (CAGD), a proper model can be achieved depending on certain characteristics of the predefined blending basis functions. The presence of these characteristics ensures the geometric properties necessary for a decent design. The objective of this study, [...] Read more.

α

-Bernstein operator in the context of its potential classification as a novel blending type basis for the construction of Bézier-like curves and surfaces. First, a recursive definition of this basis is provided, along with its unique representation in terms of that for the classical Bernstein operator. Next, following these representations, the characteristics of the basis are discussed, and one shape parameter for

α

-Bezier curves is defined. In addition, by utilizing the recursive definition of the basis, a de Casteljau-like algorithm is provided such that a subdivision schema can be applied to the construction of the new

α

-Bezier curves. The parametric continuity constraints for

C^{0}

C^{1}

, and

C^{2}

are also established to join two

α

-Bezier curves. Finally, a set of cross-sectional engineering surfaces is designed to indicate that the generalized

α

-Bernstein operator, as a basis, is efficient and easy to implement for forming shape-adjustable designs. Full article

(This article belongs to the Section Mathematics)

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14 pages, 3882 KB

Open AccessArticle

Numerical Solutions of Second-Order Elliptic Equations with C-Bézier Basis

by Lanyin Sun, Fangming Su and Kunkun Pang

Axioms 2024, 13(2), 84; https://doi.org/10.3390/axioms13020084 - 27 Jan 2024

Viewed by 1362

Abstract

This article introduces a finite element method based on the C-Bézier basis function for second-order elliptic equations. The trial function of the finite element method is set up using a combination of C-Bézier tensor product bases. One advantage of the C-Bézier basis is that it has a free shape parameter, which makes geometric modeling more convenience and flexible. The performance of the C-Bézier basis is searched for by studying three test examples. The numerical results demonstrate that this method is able to provide more accurate numerical approximations than the classical Lagrange basis. Full article

(This article belongs to the Topic Numerical Methods for Partial Differential Equations)

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19 pages, 24047 KB

Open AccessArticle

C¹-Cubic Quasi-Interpolation Splines over a CT Refinement of a Type-1 Triangulation

by Haithem Benharzallah, Abdelaziz Mennouni and Domingo Barrera

Mathematics 2023, 11(1), 59; https://doi.org/10.3390/math11010059 - 23 Dec 2022

Cited by 1 | Viewed by 2054

Abstract

C^{1}

C^{1}

continuous quasi-interpolating splines are constructed over Clough–Tocher refinement of a type-1 triangulation. Their Bernstein–Bézier coefficients are directly defined from the known values of the function to be approximated, so that a set of appropriate basis functions is not required. The resulting quasi-interpolation operators reproduce cubic polynomials. Some numerical tests are given in order to show the performance of the approximation scheme. Full article

(This article belongs to the Special Issue Computational Methods and Applications for Numerical Analysis)

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16 pages, 3642 KB

Open AccessArticle

A Comparative Study of Different Schemes Based on Bézier-like Functions with an Application of Craniofacial Fractures Reconstruction

by Abdul Majeed, Muhammad Abbas and Kenjiro T. Miura

Mathematics 2022, 10(8), 1269; https://doi.org/10.3390/math10081269 - 11 Apr 2022

Cited by 2 | Viewed by 1878

Abstract

Cranial implants, especially custom made implants, are complex, important and necessary in craniofacial fracture restoration surgery. However, the classical procedure of the manual design of the implant is time consuming and complicated. Different computer-based techniques proposed by different researchers, including CAD/CAM, mirroring, reference skull, thin plate spline and radial basis functions have been used for cranial implant restoration. Computer Aided Geometric Design (CAGD) has also been used in bio-modeling and specifically for the restoration of cranial defects in form of different spline curves, namely

C^{1}, C^{2}, G C^{1} G C^{2},

rational curves, B-spline and Non-Uniform Rational B-Spline (NURBS) curves. This paper gives an in-depth comparison of existing techniques by highlighting the limitations and advantage in different contexts. The construction of craniofacial fractures is made using different Bézier-like functions (Ball, Bernstein and Timmer basis functions) and is analyzed in detail. The

C^{1}, G C^{1}

and

G C^{2}

cubic Ball curves are performed well for construction of the small fractured part. Any form of fracture is constructed using this approach and it has been effectively applied to frontal and parietal bone fractures. However, B-spline and NURBS curves can be used for any type of fractured parts and are more friendly user. Full article

(This article belongs to the Section E1: Mathematics and Computer Science)

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7 pages, 354 KB

Open AccessArticle

Arbitrary-Order Bernstein–Bézier Functions for DGFEM Transport on 3D Polygonal Grids

by Michael Hackemack

J. Nucl. Eng. 2021, 2(3), 239-245; https://doi.org/10.3390/jne2030022 - 31 Jul 2021

Cited by 1 | Viewed by 2652

Abstract

In this paper, we present an arbitrary-order discontinuous Galerkin finite element discretization of the

S_{N}

transport equation on 3D extruded polygonal prisms. Basis functions are formed by the tensor product of 2D polygonal Bernstein–Bézier functions and 1D Lagrange polynomials. For a polynomial [...] Read more.

In this paper, we present an arbitrary-order discontinuous Galerkin finite element discretization of the

S_{N}

{x^{a} y^{b}}_{(a + b) \leq p} \otimes {z^{c}}_{c \in (0, p)}

with a dimension of

n p (p + 1) + (p + 1) (p - 1) (p - 2) / 2

on an extruded n-gon. Numerical tests confirm that the functions capture exactly monomial solutions, achieve expected convergence rates, and provide full resolution in the thick diffusion limit. Full article

(This article belongs to the Special Issue Selected Papers from PHYSOR 2020)

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25 pages, 1928 KB

Open AccessArticle

A Generalized Quasi Cubic Trigonometric Bernstein Basis Functions and Its B-Spline Form

by Yunyi Fu and Yuanpeng Zhu

Mathematics 2021, 9(10), 1154; https://doi.org/10.3390/math9101154 - 20 May 2021

Cited by 3 | Viewed by 2821

Abstract

In this paper, under the framework of Extended Chebyshev space, four new generalized quasi cubic trigonometric Bernstein basis functions with two shape functions

α (t)

and

β (t)

are constructed in a generalized quasi cubic trigonometric space span [...] Read more.

In this paper, under the framework of Extended Chebyshev space, four new generalized quasi cubic trigonometric Bernstein basis functions with two shape functions

α (t)

and

β (t)

are constructed in a generalized quasi cubic trigonometric space span

{1, \sin^{2} t, {(1 - \sin t)}^{2} α (t), {(1 - \cos t)}^{2} β (t)}

, which includes lots of previous work as special cases. Sufficient conditions concerning the two shape functions to guarantee the new construction of Bernstein basis functions are given, and three specific examples of the shape functions and the related applications are shown. The corresponding generalized quasi cubic trigonometric Bézier curves and the corner cutting algorithm are also given. Based on the new constructed generalized quasi cubic trigonometric Bernstein basis functions, a kind of new generalized quasi cubic trigonometric B-spline basis functions with two local shape functions

α_{i} (t)

and

β_{i} (t)

is also constructed in detail. Some important properties of the new generalized quasi cubic trigonometric B-spline basis functions are proven, including partition of unity, nonnegativity, linear independence, total positivity and

C^{2}

continuity. The shape of the parametric curves generated by the new proposed B-spline basis functions can be adjusted flexibly. Full article

(This article belongs to the Special Issue Computer Aided Geometric Design)

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21 pages, 776 KB

Open AccessArticle

Designing Developable C-Bézier Surface with Shape Parameters

by Caiyun Li and Chungang Zhu

Mathematics 2020, 8(3), 402; https://doi.org/10.3390/math8030402 - 11 Mar 2020

Cited by 13 | Viewed by 4003

Abstract

Developable surface plays an important role in geometric design, architectural design, and manufacturing of material. Bézier curve and surface are the main tools in the modeling of curve and surface. Since polynomial representations can not express conics exactly and have few shape handles, one may want to use rational Bézier curves and surfaces whose weights control the shape. If we vary a weight of rational Bézier curve or surface, then all of the rational basis functions will be changed. The derivation and integration of the rational curve will yield a high degree curve, which means that the shape of rational Bézier curve and surface is not easy to control. To solve this problem of shape controlling for a developable surface, we construct C-Bézier developable surfaces with some parameters using a dual geometric method. This yields properties similar to Bézier surfaces so that it is easy to design. Since C-Bézier basis functions have only two parameters in every basis, we can control the shape of the surface locally. Moreover, we derive the conditions for C-Bézier developable surface interpolating a geodesic. Full article

(This article belongs to the Special Issue Modern Geometric Modeling: Theory and Applications)

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