scholarly journals New Cubic Trigonometric Bezier-Like Functions with Shape Parameter: Curvature and Its Spiral Segment

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Abdul Majeed ◽  
Muhammad Abbas ◽  
Amna Abdul Sittar ◽  
Mohsin Kamran ◽  
Saba Tahseen ◽  
...  
Keyword(s):  
Shape Parameter ◽  
Partition Of Unity ◽  
Basis Functions ◽  
Bézier Curve ◽  
Bezier Curve ◽  
Closed Curves ◽  
Geometric Invariance ◽  
And Control ◽  
Symmetric Property ◽  
Cubic Bezier Curve

This work presents the new cubic trigonometric Bézier-type functions with shape parameter. Basis functions and the curve satisfy all properties of classical Bézier curve-like partition of unity, symmetric property, linear independent, geometric invariance, and convex hull property and have been proved. The C 3 and G 3 continuity conditions between two curve segments have also been achieved. To check the applicability of proposed functions, different types of open and closed curves have been constructed. The effect of shape parameter and control points has been observed. It is observed that, by decreasing the value of shape parameter, the curve moves toward the control polygon and vice versa. The CT-Bézier curve is closer to the cubic Bézier curve for a fixed value of shape parameter. The proposed CT-Bézier curve can be used to represent ellipse. Using proposed basis functions, we have constructed the spiral segment which is very useful to construct fair curves and desirable to design trajectories of mobile robots, highway, and railway routes’ designing.

scholarly journals An Investigation on Image Compression Using the Trigonometric Bézier Curve with a Shape Parameter

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Juncheng Li ◽  
Dongbiao Zhao
Keyword(s):  
Image Compression ◽  
Approximation Method ◽  
Shape Parameter ◽  
Bézier Curve ◽  
Bezier Curve ◽  
Control Polygon ◽  
Curve Approximation ◽  
Cubic Polynomial ◽  
The Given ◽  
Cubic Bezier Curve

A trigonometric Bézier curve analogous to the cubic polynomial Bézier curve, with a shape parameter, is presented in this work. The proposed curve inherits properties similar to those of cubic polynomial Bézier curve, and the shape of the curve can be adjusted by altering the value of the shape parameter while the control polygon is fixed. With the shape parameter, the proposed curve can be made closer to the given control polygon than the general cubic Bézier curve. Then, image compression using the trigonometric Bézier curve approximation method is investigated. Experimental results show that the trigonometric Bézier curve approximation is an effective method for dealing with image compression problems.

scholarly journals Designing Developable C-Bézier Surface with Shape Parameters

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 402 ◽  
Author(s):  
Caiyun Li ◽  
Chungang Zhu
Keyword(s):  
Architectural Design ◽  
Basis Functions ◽  
Bézier Curve ◽  
Bezier Curve ◽  
Developable Surface ◽  
Rational Bézier Curves ◽  
Polynomial Representations ◽  
Rational Bézier Curve ◽  
Shape Controlling ◽  
Two Parameters

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.

scholarly journals Generalized Fractional Bézier Curve with Shape Parameters

Mathematics ◽  
2021 ◽  
Vol 9 (17) ◽  
pp. 2141
Author(s):  
Syed Ahmad Aidil Adha Said Mad Said Mad Zain ◽  
Md Yushalify Misro ◽  
Kenjiro T. Miura
Keyword(s):  
Basis Functions ◽  
Bézier Curve ◽  
Free Form ◽  
Bezier Curve ◽  
Geometric Continuity ◽  
Shape Parameters ◽  
Complex Curves ◽  
Curves And Surfaces ◽  
New Type ◽  
Fractional Parameter

The construction of new basis functions for the Bézier or B-spline curve has been one of the most popular themes in recent studies in Computer Aided Geometric Design (CAGD). Implementing the new basis functions with shape parameters provides a different viewpoint on how new types of basis functions can develop complex curves and surfaces beyond restricted formulation. The wide selection of shape parameters allows more control over the shape of the curves and surfaces without altering their control points. However, interpolated parametric curves with higher degrees tend to overshoot in the process of curve fitting, making it difficult to control the optimal length of the curved trajectory. Thus, a new parameter needs to be created to overcome this constraint to produce free-form shapes of curves and surfaces while still preserving the basic properties of the Bézier curve. In this work, a general fractional Bézier curve with shape parameters and a fractional parameter is presented. Furthermore, parametric and geometric continuity between two generalized fractional Bézier curves is discussed in this paper, as well as demonstrating the effect of the fractional parameter of curves and surfaces. However, the conventional parametric and geometric continuity can only be applied to connect curves at the endpoints. Hence, a new type of continuity called fractional continuity is proposed to overcome this limitation. Thus, with the curve flexibility and adjustability provided by the generalized fractional Bézier curve, the construction of complex engineering curves and surfaces will be more efficient.

scholarly journals On the Involute of the Cubic Bezier Curve by Using Matrix Representation in E3

2020 ◽  
Vol 13 (2) ◽  
pp. 216-226
Author(s):  
Şeyda Kılıçoğlu ◽  
Süleyman Şenyurt
Keyword(s):  
Vector Fields ◽  
Matrix Representation ◽  
Matrix Form ◽  
Bézier Curve ◽  
Bezier Curve ◽  
Control Points ◽  
Cubic Bezier Curve

In this study we have examined, involute of the cubic Bezier curve based on the control points with matrix form in E3. Frenet vector fields and also curvatures of involute of the cubic Bezier curve are examined based on the Frenet apparatus of the first cubic Bezier curve in E3.  

Design of a Micro-Positioning Stage Using Corrugated Flexure Beam With Cubic Bézier Curve Segments

2018 ◽  
Author(s):  
Nianfeng Wang ◽  
Zhiyuan Zhang ◽  
Xianmin Zhang
Keyword(s):  
Stiffness Matrix ◽  
Matrix Method ◽  
Bézier Curve ◽  
Bezier Curve ◽  
Axial Stiffness ◽  
Control Points ◽  
Stiffness Ratio ◽  
Element Analysis ◽  
Positioning Stage ◽  
Cubic Bezier Curve

The paper introduces an analytical stiffness matrix method to model a new type of corrugated flexure (CF) beam with cubic Bézier curve segments. In order to satisfy particular design specifications, shape variation for limited geometric envelopes are often employed to alter the elastic properties of flexure hinges. In this paper, cubic Bézier curves are introduced to replace the axis of CF unit to rebuild the CF beam and the micro-positioning stage. Mohr’s integral method is applied to derive the stiffness matrix of the cubic Bézier curve segment. Modeling of the CF unit and the CF beam with cubic Bézier curve segments are further carried out through stiffness matrix method, which are confirmed by finite element analysis (FEA). Discussions about the two control points of the cubic Bézier curve segments are then conducted through search optimization, which highlights the off-axis/axial stiffness ratio and the axial compliance on the position of the two control points, to enable the micro-positioning stage both achieving high off-axis/axial stiffness ratio and large axial compliance. The derived analytical model provides a new option for the design of the CF beam.

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