scholarly journals Interpolation method for quaternionic-Bezier curves

2018 ◽  
Vol 59 ◽  
pp. 13-18
Author(s):  
Severinas Zube
Keyword(s):  
Interpolation Method ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Bézier Curve ◽  
Bezier Curve ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Three Dimensional Space ◽  
Tangent Vectors

We study rational quaternionic-Bézier curves in three dimensional space. We construct the quadratic quaternionic-Bézier curve which interpolates five points, or three points and two tangent vectors.

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 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 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.

scholarly journals Quaternionic Bézier curves, surfaces and volume

2013 ◽  
Vol 54 ◽  
Author(s):  
Severinas Zube
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Dimensional Subspace ◽  
Bezier Curves ◽  
Bézier Curves ◽  
The Real

We extended the rational Bézier construction for linear, bi-linear and threelinear map, by allowing quaternion weights. These objects are Möbius invariant and have halved degree with respect to the real parametrization. In general, these parametrizations are in four dimensional space. We analyse when a special the three-linear parametrized volume is in usual three dimensional subspace and gives three orthogonal family of Dupine cyclides.

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.

Deformation Modeling of a Flexible Instrument Using a Bézier Curve

2020 ◽  
Author(s):  
Gudala Rajesh ◽  
Jitendra P. Khatait
Keyword(s):  
Strain Energy ◽  
Dimensional Space ◽  
Three Dimensional ◽  
The Body ◽  
Geometric Constraints ◽  
Bézier Curve ◽  
Bezier Curve ◽  
Loop Formation ◽  
Surgical Interventions ◽  
Flexible Instrument

Abstract Flexible instruments are extensively used in medical applications. Manipulation of these instruments is challenging and troublesome. To study the behavior of these instruments, different modeling approaches like finite element, Cosserat rod, and differential geometry have been used. In this paper, an alternate modeling approach using Bézier curve is proposed to understand large deformation in a flexible instrument. The shape is represented by a Bézier curve. The deformation includes extension, flexure, and torsion in the instrument. Strain energy and geometric constraints are formulated using Bézier control points. The shape of the deformed instrument under the given constraints is obtained by minimizing the total strain energy. Nonlinear constrained optimization is used for minimization to find the deformed shape. The proposed method is applied for large elastic deformation in three-dimensional space. Loop formation is observed and validated with the experimental results. The proposed method will help in understanding the mechanics of a flexible instrument. Looping is a common problem during colonoscopy. The developed model will help in developing strategies for a safer introduction of these instruments inside the body for performing diagnostic and surgical interventions.

scholarly journals A Bézier Curve Based Ship Trajectory Optimization for Close-Range Maritime Operations

2017 ◽  
Author(s):  
Guoyuan Li ◽  
Houxiang Zhang
Keyword(s):  
Fuel Consumption ◽  
Trajectory Optimization ◽  
Bézier Curve ◽  
Ocean Current ◽  
Bezier Curve ◽  
Bezier Curves ◽  
Ship Maneuvering ◽  
Bézier Curves ◽  
Close Range ◽  
Maritime Operations

Ship maneuvering in close-range maritime operations is challenging for pilots, since they have to not only prevent the ship from collisions and compensate environmental impacts, but also steer it close to the target towards a proper heading. This paper presents a path planner to assist the pilots to foresee the optimal trajectory in the scenario. The path planning is formatted as an optimizing problem to minimize the turning variation fluctuation and the fuel consumption of the ship through ocean current while satisfying the constraint of orientations at the start and the end positions. Taking advantages of Bézier curves’ smoothness and adjustability, feasible trajectories are divided into two categories based on the location of the intersection between the start and end directions, and are designed as a set of parameterized Bézier curves. The variables in the Bézier curves become the state space. By searching the space using an evolutionary technique, the candidate of the Bézier curve that has the best turning and the minimized fuel consumption can be obtained. Through two case studies, the feasibility and effectiveness of the proposed planner is verified.

scholarly journals THE SERIES OF FUZZIFIED FUZZY BEZIER CURVE

2015 ◽  
Vol 78 (2-2) ◽  
Author(s):  
Rozaimi Zakaria ◽  
Abd. Fatah Wahab ◽  
R. U. Gobithaasan
Keyword(s):  
Set Theory ◽  
Fuzzy Set ◽  
Fuzzy Set Theory ◽  
Fuzzy Number ◽  
Bézier Curve ◽  
Bezier Curve ◽  
Bezier Curves ◽  
Number Concepts ◽  
Bézier Curves

This paper discusses about the series of fuzzified Bezier curves. This series of fuzzified Bezier curves constructed based on the fuzzy set theory especially fuzzy number concepts. The series of fuzzified represented by the alpha-cut of fuzzy number with various values of alpha. Then, the results were blended with Bezier curve function to produce the series of fuzzified fuzzy Bezier curves.

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