scholarly journals Geometric modeling and applications of generalized blended trigonometric Bézier curves with shape parameters

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Sidra Maqsood ◽  
Muhammad Abbas ◽  
Kenjiro T. Miura ◽  
Abdul Majeed ◽  
Azhar Iqbal
Keyword(s):  
Geometric Modeling ◽  
Recursive Formula ◽  
Basis Functions ◽  
Shape Parameters ◽  
Shape Modification ◽  
Research Topics ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Computer Aided Geometric Design ◽  
Curve Modeling

Abstract A Bézier model with shape parameters is one of the momentous research topics in geometric modeling and computer-aided geometric design. In this study, a new recursive formula in explicit expression is constructed that produces the generalized blended trigonometric Bernstein (or GBT-Bernstein, for short) polynomial functions of degree m. Using these basis functions, generalized blended trigonometric Bézier (or GBT-Bézier, for short) curves with two shape parameters are also constructed, and their geometric features and applications to curve modeling are discussed. The newly created curves share all geometric properties of Bézier curves except the shape modification property, which is superior to the classical Bézier. The $C^{3}$ C 3 and $G^{2}$ G 2 continuity conditions of two pieces of GBT-Bézier curves are also part of this study. Moreover, in contrast with Bézier curves, our generalization gives more shape adjustability in curve designing. Several examples are presented to show that the proposed method has high applied values in geometric modeling.

scholarly journals Geometric modeling of some engineering GBT-Bézier surfaces with shape parameters and their applications

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Sidra Maqsood ◽  
Muhammad Abbas ◽  
Kenjiro T. Miura ◽  
Abdul Majeed ◽  
Samia BiBi ◽  
...  
Keyword(s):  
Geometric Modeling ◽  
Basis Functions ◽  
Geometric Features ◽  
Shape Parameters ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Curves And Surfaces ◽  
Bézier Surfaces ◽  
Computer Based ◽  
Bezier Surfaces

AbstractThis study is based on some $C^{1}$ C 1 , $C^{2}$ C 2 , and $C^{3}$ C 3 continuous computer-based surfaces that are modeled by using generalized blended trigonometric Bézier (shortly, GBT-Bézier) curves with shape parameters. Initially, generalized blended trigonometric Bernstein-like (shortly, GBTB) basis functions with two shape parameters are derived in explicit expression which satisfied the basic geometric features of the traditional Bernstein basis functions. Moreover, the GBT-Bézier curves and tensor product GBT-Bézier surfaces with two shape parameters are also presented. All geometric features of the proposed GBT-Bézier curves and surfaces are similar to the traditional Bézier curves and surfaces, but the shape-adjustment is the additional feature that the traditional Bézier curves and surfaces do not hold. Finally, a class of some complex computer-based engineering surfaces via GBT-Bézier curves with shape parameters is presented. In addition, two adjacent GBT-Bézier surfaces segments are connected by higher $C^{2}$ C 2 and $C^{3}$ C 3 continuity constraints than the existing only $C^{1}$ C 1 shape adjustable Bézier surfaces. Some practical examples are provided to show the efficiency of the proposed scheme and to prove it as another powerful way for the construction and modeling of various complex composite computer-based engineering surfaces using higher-order continuities.

A novel extension to the polynomial basis functions describing Bezier curves and surfaces of degree n with multiple shape parameters

2013 ◽  
Vol 223 ◽  
pp. 1-16 ◽  
Author(s):  
Xinqiang Qin ◽  
Gang Hu ◽  
Nianjuan Zhang ◽  
Xiaoli Shen ◽  
Yang Yang
Keyword(s):  
Basis Functions ◽  
Polynomial Basis ◽  
Shape Parameters ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Curves And Surfaces

scholarly journals Geometric Modeling of Novel Generalized Hybrid Trigonometric Bézier-Like Curve with Shape Parameters and Its Applications

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 967 ◽  
Author(s):  
Samia BiBi ◽  
Muhammad Abbas ◽  
Kenjiro T. Miura ◽  
Md Yushalify Misro
Keyword(s):  
Geometric Modeling ◽  
Bézier Curve ◽  
Bezier Curve ◽  
Geometric Continuity ◽  
Shape Parameters ◽  
Bernstein Basis ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Curvature Continuity ◽  
Bernstein Basis Functions

The main objective of this paper is to construct the various shapes and font designing of curves and to describe the curvature by using parametric and geometric continuity constraints of generalized hybrid trigonometric Bézier (GHT-Bézier) curves. The GHT-Bernstein basis functions and Bézier curve with shape parameters are presented. The parametric and geometric continuity constraints for GHT-Bézier curves are constructed. The curvature continuity provides a guarantee of smoothness geometrically between curve segments. Furthermore, we present the curvature junction of complex figures and also compare it with the curvature of the classical Bézier curve and some other applications by using the proposed GHT-Bézier curves. This approach is one of the pivotal parts of construction, which is basically due to the existence of continuity conditions and different shape parameters that permit the curve to change easily and be more flexible without altering its control points. Therefore, by adjusting the values of shape parameters, the curve still preserve its characteristics and geometrical configuration. These modeling examples illustrate that our method can be easily performed, and it can also provide us an alternative strong strategy for the modeling of complex figures.

On Generalized Jacobi–Bernstein Basis Transformation: Application of Multidegree Reduction of Bézier Curves and Surfaces

2014 ◽  
Vol 14 (4) ◽  
Author(s):  
E. H. Doha ◽  
A. H. Bhrawy ◽  
M. A. Saker
Keyword(s):  
Jacobi Polynomials ◽  
Basis Functions ◽  
Least Square ◽  
Natural Basis ◽  
Bernstein Basis ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Computer Aided Geometric Design ◽  
Curves And Surfaces ◽  
Endpoint Constraints

This paper formulates a new explicit expression for the generalized Jacobi polynomials (GJPs) in terms of Bernstein basis. We also establish and prove the basis transformation between the GJPs basis and Bernstein basis and vice versa. This transformation embeds the perfect least-square performance of the GJPs with the geometrical insight of the Bernstein form. Moreover, the GJPs with indexes corresponding to the number of endpoint constraints are the natural basis functions for least-square approximation of Bézier curves and surfaces. Application to multidegree reduction (MDR) of Bézier curves and surfaces in computer aided geometric design (CAGD) is given.

scholarly journals A Novel Generalization of Trigonometric Bézier Curve and Surface with Shape Parameters and Its Applications

2020 ◽  
Vol 2020 ◽  
pp. 1-25 ◽  
Author(s):  
Sidra Maqsood ◽  
Muhammad Abbas ◽  
Gang Hu ◽  
Ahmad Lutfi Amri Ramli ◽  
Kenjiro T. Miura
Keyword(s):  
Basis Functions ◽  
Geometric Continuity ◽  
Shape Parameters ◽  
Bernstein Basis ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Surface Design ◽  
Curves And Surfaces ◽  
Bernstein Basis Functions ◽  
Parametric Continuity

Adopting a recurrence technique, generalized trigonometric basis (or GT-basis, for short) functions along with two shape parameters are formulated in this paper. These basis functions carry a lot of geometric features of classical Bernstein basis functions and maintain the shape of the curve and surface as well. The generalized trigonometric Bézier (or GT-Bézier, for short) curves and surfaces are defined on these basis functions and also analyze their geometric properties which are analogous to classical Bézier curves and surfaces. This analysis shows that the existence of shape parameters brings a convenience to adjust the shape of the curve and surface by simply modifying their values. These GT-Bézier curves meet the conditions required for parametric continuity (C0, C1, C2, and C3) as well as for geometric continuity (G0, G1, and G2). Furthermore, some curve and surface design applications have been discussed. The demonstrating examples clarify that the new curves and surfaces provide a flexible approach and mathematical sketch of Bézier curves and surfaces which make them a treasured way for the project of curve and surface modeling.

scholarly journals Curve and surface construction based on the generalized toric-Bernstein basis functions

2020 ◽  
Vol 18 (1) ◽  
pp. 36-56 ◽  
Author(s):  
Jing-Gai Li ◽  
Chun-Gang Zhu
Keyword(s):  
Computer Aided Design ◽  
Geometric Modeling ◽  
Basis Functions ◽  
Bernstein Basis ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Parametric Curve ◽  
Curves And Surfaces ◽  
Computer Aided ◽  
Bernstein Basis Functions

Abstract The construction of parametric curve and surface plays an important role in computer aided geometric design (CAGD), computer aided design (CAD), and geometric modeling. In this paper, we define a new kind of blending functions associated with a real points set, called generalized toric-Bernstein (GT-Bernstein) basis functions. Then, the generalized toric-Bézier (GT-Bézier) curves and surfaces are constructed based on the GT-Bernstein basis functions, which are the projections of the (irrational) toric varieties in fact and the generalizations of the classical rational Bézier curves/surfaces and toric surface patches. Furthermore, we also study the properties of the presented curves and surfaces, including the limiting properties of weights and knots. Some representative examples verify the properties and results.

scholarly journals Quasi-Bézier Curves with Shape Parameters

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Jun Chen
Keyword(s):  
Convex Hull ◽  
Basis Functions ◽  
Geometrical Shape ◽  
Control Points ◽  
Shape Parameters ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Numerical Examples ◽  
Universal Form ◽  
Special Cases

The universal form of univariate Quasi-Bézier basis functions with multiple shape parameters and a series of corresponding Quasi-Bézier curves were constructed step-by-step in this paper, using the method of undetermined coefficients. The series of Quasi-Bézier curves had geometric and affine invariability, convex hull property, symmetry, interpolation at the endpoints and tangent edges at the endpoints, and shape adjustability while maintaining the control points. Various existing Quasi-Bézier curves became special cases in the series. The obvious geometric significance of shape parameters made the adjustment of the geometrical shape easier for the designer. The numerical examples indicated that the algorithm was valid and can easily be applied.

New method of curve modeling based on continuity for extension-Bézier curves

2008 ◽  
Vol 28 (1) ◽  
pp. 187-190
Author(s):  
Gang HU
Keyword(s):  
New Method ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Curve Modeling

A Class of Generalized Bézier Curves and Surfaces with Multiple Shape Parameters

2010 ◽  
Vol 22 (5) ◽  
pp. 838-844 ◽  
Author(s):  
Zhi Liu ◽  
Xiaoyan Chen ◽  
Ping Jiang
Keyword(s):  
Shape Parameters ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Curves And Surfaces
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