scholarly journals Approximate Multidegree Reduction ofλ-Bézier Curves

2016 ◽  
Vol 2016 ◽  
pp. 1-12 ◽  
Author(s):  
Gang Hu ◽  
Huanxin Cao ◽  
Suxia Zhang
Keyword(s):  
Shape Control ◽  
Control Parameter ◽  
Linear Equations ◽  
Least Square ◽  
Bézier Curve ◽  
Bezier Curve ◽  
Control Points ◽  
Degree Reduction ◽  
Bezier Curves ◽  
Bézier Curves

Besides inheriting the properties of classical Bézier curves of degreen, the correspondingλ-Bézier curves have a good performance in adjusting their shapes by changing shape control parameter. In this paper, we derive an approximation algorithm for multidegree reduction ofλ-Bézier curves in theL2-norm. By analysing the properties ofλ-Bézier curves of degreen, a method which can deal with approximatingλ-Bézier curve of degreen+1byλ-Bézier curve of degreem  (m≤n)is presented. Then, in unrestricted andC0,C1constraint conditions, the new control points of approximatingλ-Bézier curve can be obtained by solving linear equations, which can minimize the least square error between the approximating curves and the original ones. Finally, several numerical examples of degree reduction are given and the errors are computed in three conditions. The results indicate that the proposed method is effective and easy to implement.

A Bezier Curve Based Key Management Scheme for Hierarchical Wireless Sensor Networks

2012 ◽  
Vol 263-266 ◽  
pp. 2979-2985
Author(s):  
Yong Luo Shen ◽  
Jun Zhang ◽  
Di Wei Yang ◽  
Lin Bo Luo
Keyword(s):  
Key Management ◽  
Wireless Sensor ◽  
Bézier Curve ◽  
Bezier Curve ◽  
Control Points ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Management Scheme ◽  
Secret Keys ◽  
Key Management Scheme

In this paper, we propose a novel key management scheme based on Bezier curves for hierarchical wireless sensor networks (WSNs). The design of our scheme is motivated by the idea that a Bezier curve can be subdivided into arbitrarily continuous pieces of sub Bezier curves. The subdivided sub Bezier curves are easily organized to a hierarchical architecture that is similar to hierarchical WSNs. The subdivided Bezier curves are unique and independent from each other so that it is suitable to assign each node in the WSN with a sub Bezier curve. Since a piece of Bezier curve can be presented by its control points, in the proposed key management scheme, the secret keys for each node are selected from the corresponding Bezier curve’s control points. Comparing with existing key management schemes, the proposed scheme is more suitable for distributing secret keys for hierarchical WSNs and more efficient in terms of computational and storage cost.

Multi-Degree Reduction of DP Curves with Constraints of Endpoints Continuity

2012 ◽  
Vol 215-216 ◽  
pp. 669-673
Author(s):  
Hua Hui Cai ◽  
Yan Cheng ◽  
Yong Hong Zhu
Keyword(s):  
Bézier Curve ◽  
Bezier Curve ◽  
Reduction Algorithm ◽  
Degree Reduction ◽  
Bernstein Basis ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Conversion Formula

In this paper, we presented a constrained multi-degree reduction algorithm of DP curves based on the transformation between the DP and Bézier curves. We first correct the conversion formula between Bernstein basis and DP basis. And then, we deal with multi-degree reduction of NP curves by degree reduction of Bézier curve.

scholarly journals Shape Modification forλ-Bézier Curves Based on Constrained Optimization of Position and Tangent Vector

2015 ◽  
Vol 2015 ◽  
pp. 1-12 ◽  
Author(s):  
Gang Hu ◽  
Xiaomin Ji ◽  
Xinqiang Qin ◽  
Suxia Zhang
Keyword(s):  
Constrained Optimization ◽  
Shape Control ◽  
Control Parameter ◽  
Tangent Vector ◽  
Control Points ◽  
Shape Modification ◽  
Shape Preserving ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Curve Design

Besides inheriting the properties of classical Bézier curves of degreen, the correspondingλ-Bézier curves have a good performance on adjusting their shapes by changing shape control parameter. Specially, in the case where the shape control parameter equals zero, theλ-Bézier curves degenerate to the classical Bézier curves. In this paper, the shape modification ofλ-Bézier curves by constrained optimization of position and tangent vector is investigated. The definition and properties ofλ-Bézier curves are given in detail, and the shape modification is implemented by optimizing perturbations of control points. At the same time, the explicit formulas of modifying control points and shape parameter are obtained by Lagrange multiplier method. Using this algorithm,λ-Bézier curves are modified to satisfy the specified constraints of position and tangent vector, meanwhile the shape-preserving property is still retained. In order to illustrate its ability on adjusting the shape ofλ-Bézier curves, some curve design applications are discussed, which show that the proposed method is effective and easy to implement.

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.

scholarly journals Approximate conversion of Bézier curves

1995 ◽  
Vol 51 (1) ◽  
pp. 153-162 ◽  
Author(s):  
Yungeom Park ◽  
U Jin Choi ◽  
Ha-Jine Kimn
Keyword(s):  
Error Analysis ◽  
A Priori ◽  
Approximation Error ◽  
Bézier Curve ◽  
Bezier Curve ◽  
A Priori Bounds ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Geometric Properties

The methods for generating a polynomial Bézier approximation of degree n − 1 to an nth degree Bézier curve, and error analysis, are presented. The methods are based on observations of the geometric properties of Bézier curves. The approximation agrees at the two endpoints up to a preselected smoothness order. The methods allow a detailed error analysis, providing a priori bounds of the point-wise approximation error. The error analysis for other authors’ methods is also presented.

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