scholarly journals G3 Shape Adjustable GHT-Bézier Developable Surfaces and Their Applications

Mathematics ◽  
2021 ◽  
Vol 9 (19) ◽  
pp. 2350
Author(s):  
Samia BiBi ◽  
Md Yushalify Misro ◽  
Muhammad Abbas ◽  
Abdul Majeed ◽  
Tahir Nazir
Keyword(s):  
Shape Control ◽  
Duality Principle ◽  
Developable Surface ◽  
Control Parameters ◽  
Shape Parameters ◽  
Developable Surfaces ◽  
Bézier Surfaces ◽  
Canal Surfaces ◽  
Novel Method ◽  
Bezier Surfaces

In this article, we proposed a novel method for the construction of generalized hybrid trigonometric (GHT-Bézier) developable surfaces to tackle the issue of modeling and shape designing in engineering. The GHT-Bézier developable surface is obtained by using the duality principle between the points and planes with GHT-Bézier curve. With different shape control parameters in their domain, a class of GHT-Bézier developable surfaces can be established (such as enveloping developable GHT-Bézier surfaces, spine curve developable GHT-Bézier surfaces, geodesic interpolating surfaces for GHT-Bézier surface and developable GHT-Bézier canal surfaces), which possess many properties of GHT-Bézier surfaces. By changing the values of shape parameters the effect on the developable surface is obvious. In addition, some useful geometric properties of GHT-Bézier developable surface and the G1, G2 (Farin-Boehm and Beta) and G3 continuity conditions between any two GHT-Bézier developable surfaces are derived. Furthermore, various useful and representative numerical examples demonstrate the convenience and efficiency of the proposed method.

scholarly journals Shape-adjustable developable generalized blended trigonometric Bézier surfaces and their applications

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Sidra Maqsood ◽  
Muhammad Abbas ◽  
Kenjiro T. Miura ◽  
Abdul Majeed ◽  
Gang Hu ◽  
...  
Keyword(s):  
Computer Aided Design ◽  
Geometric Modeling ◽  
Architectural Design ◽  
Duality Principle ◽  
Shape Parameters ◽  
Developable Surfaces ◽  
Bézier Surfaces ◽  
Current Curve ◽  
Bezier Surfaces ◽  
Necessary And Sufficient

AbstractDevelopable surfaces have a vital part in geometric modeling, architectural design, and material manufacturing. Developable Bézier surfaces are the important tools in the construction of developable surfaces, but due to polynomial depiction and having no shape parameter, they cannot describe conics exactly and can only handle a few shapes. To tackle these issues, two straightforward techniques are proposed to the computer-aided design of developable generalized blended trigonometric Bézier surfaces (for short, developable GBT-Bézier surfaces) with shape parameters. A developable GBT-Bézier surface is established by making a collection of control planes with generalized blended trigonometric Bernstein-like (for short, GBTB) basis functions on duality principle among points and planes in 4D projective space. By changing the values of shape parameters, a group of developable GBT-Bézier surfaces that preserves the features of the developable GBT-Bézier surfaces can be generated. Furthermore, for a continuous connection among these developable GBT-Bézier surfaces, the necessary and sufficient $G^{1}$ G 1 and $G^{2}$ G 2 (Farin–Boehm and beta) continuity conditions are also defined. Some geometric designs of developable GBT-Bézier surfaces are illustrated to show that the suggested scheme can settle the shape and position adjustment problem of developable Bézier surfaces in a better way than other existing schemes. Hence, the suggested scheme has not only all geometric features of current curve design schemes but surpasses their imperfections which are usually faced in engineering.

Geometric Design and Fabrication of Developable Bezier Surfaces

1992 ◽  
Author(s):  
R. M. C. Bodduluri ◽  
B. Ravani
Keyword(s):  
Geometric Design ◽  
Developable Surface ◽  
Developable Surfaces ◽  
Surface Patches ◽  
Bézier Surfaces ◽  
Point Representation ◽  
Cad Cam ◽  
C2 Continuity ◽  
Type Construction ◽  
Bezier Surfaces

Abstract In this paper we study Computer Aided Geometric Design (CAGD) and Manufacturing (CAM) of developable surfaces. We develop direct representations of developable surfaces in terms of point as well as plane geometries. The point representation uses a Bezier curve, the tangents of which span the surface. The plane representation uses control planes instead of control points and determines a surface which is a Bezier interpolation of the control planes. In this case, a de Casteljau type construction method is presented for geometric design of developable Bezier surfaces. In design of piecewise surface patches, a computational geometric algorithm similar to Farin-Boehm construction used in design of piecewise parametric curves is developed for designing developable surfaces with C2 continuity. In the area of manufacturing or fabrication of developable surfaces, we present simple methods for both development of a surface into a plane and bending of a flat plane into a desired developable surface. The approach presented uses plane and line geometries and eliminates the need for solving differential equations of Riccatti type used in previous methods. The results are illustrated using an example generated by a CAD/CAM system implemented based on the theory presented.

Construction of generalized developable Bézier surfaces with shape parameters

2018 ◽  
Vol 41 (17) ◽  
pp. 7804-7829 ◽  
Author(s):  
Gang Hu ◽  
Huanxin Cao ◽  
Xinqiang Qin
Keyword(s):  
Shape Parameters ◽  
Bézier Surfaces ◽  
Bezier Surfaces

scholarly journals Generalized Developable Cubic Trigonometric Bézier Surfaces

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 283
Author(s):  
Muhammad Ammad ◽  
Md Yushalify Misro ◽  
Muhammad Abbas ◽  
Abdul Majeed
Keyword(s):  
Shape Parameter ◽  
Surface Characteristics ◽  
Surface Shape ◽  
Shape Parameters ◽  
New Approach ◽  
Bézier Surface ◽  
Bezier Surface ◽  
Bézier Surfaces ◽  
Bezier Surfaces ◽  
Shape Adjustment

This paper introduces a new approach for the fabrication of generalized developable cubic trigonometric Bézier (GDCT-Bézier) surfaces with shape parameters to address the fundamental issue of local surface shape adjustment. The GDCT-Bézier surfaces are made by means of GDCT-Bézier-basis-function-based control planes and alter their shape by modifying the shape parameter value. The GDCT-Bézier surfaces are designed by maintaining the classic Bézier surface characteristics when the shape parameters take on different values. In addition, the terms are defined for creating a geodesic interpolating surface for the GDCT-Bézier surface. The conditions appropriate and suitable for G1, Farin–Boehm G2, and G2 Beta continuity in two adjacent GDCT-Bézier surfaces are also created. Finally, a few important aspects of the newly formed surfaces and the influence of the shape parameters are discussed. The modeling example shows that the proposed approach succeeds and can also significantly improve the capability of solving problems in design engineering.

scholarly journals A novel method for planning liver resections using deformable Bézier surfaces and distance maps

2017 ◽  
Vol 144 ◽  
pp. 135-145 ◽  
Author(s):  
Rafael Palomar ◽  
Faouzi A. Cheikh ◽  
Bjørn Edwin ◽  
Åsmund Fretland ◽  
Azeddine Beghdadi ◽  
...  
Keyword(s):  
Liver Resections ◽  
Bézier Surfaces ◽  
Novel Method ◽  
Bezier Surfaces

Shape control of Bézier surfaces with iso-parametric monotone curvature constraints

2003 ◽  
Vol 20 (6) ◽  
pp. 383-394 ◽  
Author(s):  
Yulin Wang ◽  
Shuchun Wang ◽  
Luzou Zhang ◽  
Bingyan Zhao
Keyword(s):  
Shape Control ◽  
Bézier Surfaces ◽  
Monotone Curvature ◽  
Bezier Surfaces

Cubic Uniform B-Spline Curve and Surface with Multiple Shape Parameters

2014 ◽  
Vol 543-547 ◽  
pp. 1860-1863
Author(s):  
Xi Wang ◽  
Cui Cui Gao ◽  
Chen Jiang
Keyword(s):  
Shape Control ◽  
Basis Functions ◽  
Control Parameters ◽  
Shape Parameters ◽  
Surface Design ◽  
Local Shape ◽  
B Spline ◽  
Spline Curves ◽  
Curve Fit ◽  
Polynomial Curve

In order to construct B-spline curves with local shape control parameters, a class of polynomial basis functions with two local shape parameters is presented. Properties of the proposed basis functions are analyzed and the corresponding piecewise polynomial curve is constructed with two local shape control parameters accordingly. In particular, the G1 continuous and the shapes of other segments of the curve can remain unchangeably during the manipulation on the shape of each segment on the curve. Numerical examples illustrate that the constructed curve fit to the control polygon very well. Furthermore, its applications in curve design is discussed and an extend application on surface design is also presented. Modeling examples show that the new curve is very valuable for the design of curves and surfaces.

scholarly journals A Class of Sextic Trigonometric Bézier Curve with Two Shape Parameters

2021 ◽  
Vol 2021 ◽  
pp. 1-16
Author(s):  
Salma Naseer ◽  
Muhammad Abbas ◽  
Homan Emadifar ◽  
Samia Bi Bi ◽  
Tahir Nazir ◽  
...  
Keyword(s):  
Shape Control ◽  
Basis Functions ◽  
Control Parameters ◽  
Shape Parameters ◽  
Bernstein Basis ◽  
Closed Curves ◽  
Geometric Characteristics ◽  
New Class ◽  
Bernstein Basis Functions ◽  
Short Basis

In this paper, we present a new class of sextic trigonometric Bernstein (ST-Bernstein, for short) basis functions with two shape parameters along with their geometric properties which are similar to the classical Bernstein basis functions. A sextic trigonometric Bézier (ST-Bézier, for short) curve with two shape parameters and their geometric characteristics is also constructed. The continuity constraints for the connection of two adjacent ST-Bézier curves segments are discussed. Shape control parameters can provide an opportunity to modify the shape of curve as designer desired. Some open and closed curves are also part of this study.

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.

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