scholarly journals Degree reduction of Bezier curves and its error analysis

1995 ◽  
Vol 36 (4) ◽  
pp. 399-413 ◽  
Author(s):  
Yunbeom Park ◽  
U Jin Choi
Keyword(s):  
Error Analysis ◽  
A Priori ◽  
Error Tolerance ◽  
Bézier Curve ◽  
Reduction Algorithm ◽  
Degree Reduction ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Geometric Properties ◽  
Original Curve

AbstractThe error analysis of an algorithm for generating an approximation of degree n − 1 to an nth degree Bézier curve is presented. The algorithm is based on observations of the geometric properties of Bézier curves which allow the development of detailed error analysis. By combining subdivision with a degree reduction algorithm, a piecewise approximation can be generated, which is within some preset error tolerance of the original curve. The number of subdivisions required can be determined a priori and a piecewise approximation of degree m can be generated by iterating the scheme.

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.

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 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.

scholarly journals Approximate Multi-Degree Reduction of SG-Bézier Curves Using the Grey Wolf Optimizer Algorithm

Symmetry ◽  
2019 ◽  
Vol 11 (10) ◽  
pp. 1242
Author(s):  
Hu ◽  
Qiao ◽  
Qin ◽  
Wei
Keyword(s):  
Computer Aided Design ◽  
Fitness Function ◽  
Grey Wolf Optimizer ◽  
Grey Wolf ◽  
Degree Reduction ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Geometric Properties ◽  
Reduction Effect ◽  
Aided Design

SG-Bézier curves have become a useful tool for shape design and geometric representation in computer aided design (CAD), owed to their good geometric properties, e.g., symmetry and convex hull property. Aiming at the problem of approximate degree reduction of SG-Bézier curves, a method is proposed to reduce the n-th SG-Bézier curves to m-th (m < n) SG-Bézier curves. Starting from the idea of grey wolf optimizer (GWO) and combining the geometric properties of SG-Bézier curves, this method converts the problem of multi-degree reduction of SG-Bézier curves into solving an optimization problem. By choosing the fitness function, the approximate multi-degree reduction of SG-Bézier curves with adjustable shape parameters is realized under unrestricted and corner interpolation constraints. At the same time, some concrete examples of degree reduction and its errors are given. The results show that this method not only achieves good degree reduction effect, but is also easy to implement and has high accuracy.

scholarly journals Degree Reduction of Q-Bézier Curves via Squirrel Search Algorithm

Mathematics ◽  
2021 ◽  
Vol 9 (18) ◽  
pp. 2212
Author(s):  
Xiaomin Liu ◽  
Muhammad Abbas ◽  
Gang Hu ◽  
Samia BiBi
Keyword(s):  
Objective Function ◽  
Search Algorithm ◽  
Shape Design ◽  
Degree Reduction ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Good Shape ◽  
Original Curve ◽  
Reduction Effect ◽  
Optimal Set

Q-Bézier curves find extensive applications in shape design owing to their excellent geometric properties and good shape adjustability. In this article, a new method for the multiple-degree reduction of Q-Bézier curves by incorporating the swarm intelligence-based squirrel search algorithm (SSA) is proposed. We formulate the degree reduction as an optimization problem, in which the objective function is defined as the distance between the original curve and the approximate curve. By using the squirrel search algorithm, we search within a reasonable range for the optimal set of control points of the approximate curve to minimize the objective function. As a result, the optimal approximating Q-Bézier curve of lower degree can be found. The feasibility of the method is verified by several examples, which show that the method is easy to implement, and good degree reduction effect can be achieved using it.

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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