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.

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 A Class of Trigonometric Bernstein-Type Basis Functions with Four Shape Parameters

2019 ◽  
Vol 2019 ◽  
pp. 1-16 ◽  
Author(s):  
Yuanpeng Zhu ◽  
Zhuo Liu
Keyword(s):  
Total Positivity ◽  
Basis Functions ◽  
Tension Control ◽  
Control Points ◽  
Shape Parameters ◽  
Bernstein Type ◽  
Bézier Curves ◽  
Type Curves ◽  
Corner Cutting ◽  
Cutting Algorithm

In this work, a family of four new trigonometric Bernstein-type basis functions with four shape parameters is constructed, which form a normalized basis with optimal total positivity. Based on the new basis functions, a kind of trigonometric Bézier-type curves with four shape parameters, analogous to the cubic Bézier curves, is constructed. With appropriate choices of control points and shape parameters, the resulting trigonometric Bézier-type curves can represent exactly any arc of an ellipse or parabola. The four shape parameters have tension control roles on adjusting the shape of resulting curves. Moreover, a new corner cutting algorithm is also proposed for calculating the trigonometric Bézier-type curves stably and efficiently.

scholarly journals Numerical Solution for IVP in Volterra Type Linear Integrodifferential Equations System

2013 ◽  
Vol 2013 ◽  
pp. 1-4 ◽  
Author(s):  
F. Ghomanjani ◽  
A. Kılıçman ◽  
S. Effati
Keyword(s):  
Numerical Solution ◽  
Integrodifferential Equations ◽  
Control Points ◽  
Volterra Type ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Numerical Examples ◽  
Piecewise Polynomials

A method is proposed to determine the numerical solution of system of linear Volterra integrodifferential equations (IDEs) by using Bezier curves. The Bezier curves are chosen as piecewise polynomials of degreen, and Bezier curves are determined on[t0, tf ]by n+1control points. The efficiency and applicability of the presented method are illustrated by some numerical examples.

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 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.

scholarly journals Bezier Curves for Solving Fredholm Integral Equations of the Second Kind

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
F. Ghomanjani ◽  
M. H. Farahi ◽  
A. Kılıçman
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Fredholm Integral Equation ◽  
Fredholm Integral Equations ◽  
Direct Algorithm ◽  
Control Points ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Numerical Examples ◽  
Piecewise Polynomials

The Bezier curves are presented to estimate the solution of the linear Fredholm integral equation of the second kind. A direct algorithm for solving this problem is given. We have chosen the Bezier curves as piecewise polynomials of degreenand determine Bezier curves on [0, 1] byn+1control points. Numerical examples illustrate that the algorithm is applicable and very easy to use.

Smoothness of C-Bezier Curves

2000 ◽  
Author(s):  
Manhong Wen ◽  
Kwun-Lon Ting
Keyword(s):  
Shape Control ◽  
Control Point ◽  
Design Procedure ◽  
Design Parameters ◽  
Control Points ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Point Distribution ◽  
G1 Continuity ◽  
G2 Continuity

Abstract This paper presents G1 and G2 continuity conditions of c-Bezier curves. It shows that the collinear condition for G1 continuity of Bezier curves is generally no longer necessary for c-Bezier curves. Such a relaxation of constraints on control points is beneficial from the structure of c-Bezier curves. By using vector weights, each control point has two extra free design parameters, which offer the probability of obtaining G1 and G2 continuity by only adjusting the weights if the control points are properly distributed. The enlargement of control point distribution region greatly simplifies the design procedure to and enhances the shape control on constructing composite curves.

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.

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