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 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 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 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 Using Bezier curves in medical applications

Filomat ◽  
2016 ◽  
Vol 30 (4) ◽  
pp. 937-943 ◽  
Author(s):  
Buket Simsek ◽  
Ahmet Yardimci
Keyword(s):  
Cross Sections ◽  
Biomedical Applications ◽  
Medical Physics ◽  
Binomial Coefficients ◽  
Medical Applications ◽  
Bernstein Basis ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Combinatorial Sums ◽  
Bernstein Basis Functions

In this paper we survey the 3D reconstruction of an object from its 2D cross-sections has many applications in different fields of sciences such as medical physics and biomedical applications. The aim of this paper is to give not only the Bezier curves in medical applications, but also by using generating functions for the Bernstein basis functions and their identities, some combinatorial sums involving binomial coefficients are deriven. Finally, we give some comments related to the above areas.

scholarly journals Explicit Continuity Conditions for G1 Connection of S-λ Curves and Surfaces

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1359 ◽  
Author(s):  
Gang Hu ◽  
Huinan Li ◽  
Muhammad Abbas ◽  
Kenjiro T. Miura ◽  
Guoling Wei
Keyword(s):  
Computer Aided Design ◽  
Linear Independence ◽  
Basis Functions ◽  
Geometric Continuity ◽  
Complex Curves ◽  
Curves And Surfaces ◽  
Computer Aided ◽  
Cad Cam ◽  
Geometric Representations ◽  
Aided Design

The S-λ model is one of the most useful tools for shape designs and geometric representations in computer-aided geometric design (CAGD), which is due to its good geometric properties such as symmetry, shape adjustable property. With the aim to solve the problem that complex S-λ curves and surfaces cannot be constructed by a single curve and surface, the explicit continuity conditions for G1 connection of S-λ curves and surfaces are investigated in this paper. On the basis of linear independence and terminal properties of S-λ basis functions, the conditions of G1 geometric continuity between two adjacent S-λ curves and surfaces are proposed, respectively. Modeling examples imply that the continuity conditions proposed in this paper are easy and effective, which indicate that the S-λ curves and surfaces can be used as a powerful supplement of complex curves and surfaces design in computer aided design/computer aided manufacturing (CAD/CAM) system.

scholarly journals Geometric Modeling of the Valencia Orange (Citrus sinensis L.) by Applying Bézier Curves and an Image-Based CAD Approach

Agriculture ◽  
2020 ◽  
Vol 10 (8) ◽  
pp. 313
Author(s):  
Hector A. Tinoco ◽  
Daniel R. Barco ◽  
Olga Ocampo ◽  
Jaime Buitrago-Osorio
Keyword(s):  
Computer Aided Design ◽  
Citrus Sinensis ◽  
Geometric Modeling ◽  
Prediction Models ◽  
Two Dimensions ◽  
Radius Of Curvature ◽  
Computer Design ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Valencia Orange

The computer-aided design of fruits are used for different purposes, e.g., to determine mechanical properties by applying engineering simulations, to design postharvest equipment, and to study the natural changes related to the topology. This paper developed a methodology to model Valencia orange (Citrus sinensis), applying Bézier curves and an image-based CAD approach; the orange geometry was designed for different ripening stages. In the modeling process, a 3D construction was carried out using third-order Bézier curves, adjusted to the images taken in orthogonal planes. Four control points defined each profile to compose the geometric pattern of the orange, with geometric errors lower than 3%. Two prediction models were proposed to relate the orthogonal dimensions with a factor size; this means that two dimensions out of three can be predicted. The results showed that the shape ratios kept constant in any ripening stage; however, the radius of curvature evidenced differences in the analyzed shape profiles. The methodological framework presented in the paper might be used to draw other types of citrus fruits. This contribution is a tool to model fruits in 3D, instead of using expensive technological equipment, since it is only necessary to apply computer design tools.

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.

Regularity of Bézier Curves

2011 ◽  
Vol 48-49 ◽  
pp. 877-880
Author(s):  
Qing Xian Meng ◽  
Hui Li Liu
Keyword(s):  
Computer Aided Design ◽  
Practical Method ◽  
Singular Points ◽  
Algebraic Equations ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Computer Aided ◽  
Aided Design ◽  
Zero Points ◽  
Isolation Theorem

Regularity is one of important properties of curves in computer aided design. In this paper, we convert the problem of determining regularity of Bézier curves to that of detecting existence of zero points of polynomials. Based on the properties of algebraic equations and isolation theorem of roots, a simple and practical method is presented. Regularity of Bézier curves and number of singular points can be determined by easier computation.

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