Some remarks on Bézier curves

Parametric polynomial curves in Bézier-Bernstein representation are considered as prohections of rational norm curves of degree n in n-space; from this point of view the singularities of a planar Bézier cubic are determined and expressed by its affine invariants. Secondly, for an arbitrary pair of adjacent parametric curves in homogeneous coordinates, the general conditions for geometric continuity of any order k, G^k, are established.This result generalizes the corresponding conditions in the non-homogeneous (affine) case, recently obtained by [Goodman '84]. Some applications are given for Bézier curves. In particular, for γ-splines [Boehm '85], the existence of a global rational parameter that makes it to a C² parametric curve is shown. Furthermore, for two adjacent rational Bézier curves the complete conditions for G³ are stated using the projective properties of the control points only.     

Self-intersections of rational Bézier curves

Rational Bézier curves provide a curve fitting tool and are widely used in Computer Aided Geometric Design, Computer Aided Design and Geometric Modeling. The injectivity (one-to-one property) of rational Bézier curve as a mapping function is equivalent to the curve without self-intersections. We present a geometric condition on the control polygon which is equivalent to the injectivity of rational Bézier curve with this control polygon for all possible choices of weights. The proof is based on the degree elevation and toric degeneration of rational Bézier curve.     

Geometric construction of energy-minimizing Bézier curves

Modeling energy-minimizing curves have many applications and are a basic problem of Geometric Modeling.In this paper,we propose the method for geometric design of energy-minimizing B′ezier curves.Firstly,the necessary and sufficient condition on the control points for B′ezier curves to have minimal internal energy is derived.Based on this condition,we propose the geometric constructions of three kinds of B′ezier curves with minimal internal energy including stretch energy,strain energy and jerk energy.Given...     

Bilinear precision of rational Bézier surfaces

This short note proves the bilinear precision property of rational Bézier surfaces and gives a simple algorithm to compute the weights. Linear precision of rational Bézier surfaces is also discussed.     

Partial derivatives of rational Bézier surfaces

A rational surface is the locus of a rational curve that is moving through space and thereby changing its shape by changing its control points and weights. This intuitive definition can be used to derive hodographs of rational Bézier surfaces and their bounds of magnitude.     

Optimal cooperative collision avoidance between multiple robots based on Bernstein–Bézier curves

In this paper a new cooperative collision-avoidance method for multiple, nonholonomic robots based on Bernstein–Bézier curves is presented. The main contribution focuses on an optimal, cooperative, collision avoidance for a multi-robot system where the velocities and accelerations of the mobile robots are constrained and the start and the goal velocity are defined for each robot. The optimal path of each robot, from the start pose to the goal pose, is obtained by minimizing the penalty function, which takes into account the sum of all the path lengths subjected to the distances between the robots, which should be larger than the minimum distance defined as the safety distance, and subjected to the velocities and accelerations, which should be lower than the maximum allowed for each robot. The model-predictive trajectory tracking is used to drive the robots on the obtained reference paths. The results of the path planning, real experiments and some future work ideas are discussed.     

Corner cutting curves and a new characterization of Bézier and B-spline curves

We call a curve c corner cutting, if each single point of c can be received from a given polygon by a finite corner cutting process. We prove some results about these curves, especially about interpolation and contact. The obtained results will give a new geometric characterization of Bézier and B-spline curves.     

Low-harmonic rational Bézier curves for trajectory generation of high-speed machinery

For a given high-speed machinery, a significant source of the internally induced vibrational excitation is the presence of high frequency harmonics in the trajectories that the system is forced to follow. In this paper a special class of rational Bézier curves is presented that correspond to low-harmonic trajectory patterns. Harmonic Bernstein polynomials and harmonic deCasteljau algorithm are also introduced as two major tools for generating the harmonic rational Bézier curves. These curves can be used to synthesize trajectories within the dynamic response limitations of the actuators while avoiding the excitation of the natural modes of vibration of the system.     

Degree reduction of Bézier curves with restricted control points area

In this paper, we propose a new approach to the problem of degree reduction of Bézier curves with respect to the least squares norm. We impose restrictions of the control points area to get more intuitive location of the control points. As a result, the degree reduced curve is suitable for further modification and applications. We present and compare two methods of solving the new problem. First one is based on quadratic programming approach and the other on BVLS algorithm. Some illustrative examples are given.     

C 1 NURBS representations of G 1 composite rational Bézier curves

This paper is concerned with the re-representation of a G ¹ composite rational Bézier curve. Although the rational Bézier curve segments that form the composite curve are G ¹ continuous at their joint points, their homogeneous representations may not be even C ⁰ continuous in the homogeneous space. In this paper, an algorithm is presented to convert the G ¹ composite rational Bézier curve into a NURBS curve whose nonrational homogeneous representation is C ¹ continuous in the homogeneous space. This re-representation process involves reparameterization using Möbius transformations, smoothing multiplication and parameter scaling transformations. While the previous methods may fail in some situations, the method proposed in this paper always works.     

Multi-degree reduction of tensor product Bézier surfaces with conditions of corners interpolations

This paper studies the multi-degree reduction of tensor product B(?)zier surfaces with any degree interpolation conditions of four corners, which is urgently to be resolved in many CAD/CAM systems. For the given conditions of corners interpolation, this paper presents one intuitive method of degree reduction of parametric surfaces. Another new approximation algorithm of multi-degree reduction is also presented with the degree elevation of surfaces and the Chebyshev polynomial approximation theory. It obtains the good approximate effect and the boundaries of degree reduced surface can preserve the prescribed continuities. The degree reduction error of the latter algorithm is much smaller than that of the first algorithm. The error bounds of degree reduction of two algorithms are also presented .     

On the smooth convergence of subdivision and degree elevation for Bézier curves

Bézier subdivision and degree elevation algorithms generate piecewise linear approximations of Bézier curves that converge to the original Bézier curve. Discrete derivatives of arbitrary order can be associated with these piecewise linear functions via divided differences. Here we establish the convergence of these discrete derivatives to the corresponding continuous derivatives of the initial Bézier curve. Thus, we show that the control polygons generated by subdivision and degree elevation provide not only an approximation to a Bézier curve, but also approximations of its derivatives of arbitrary order.     

Lagrange geometric interpolation by rational spatial cubic Bézier curves

In the paper, the Lagrange geometric interpolation by spatial rational cubic Bézier curves is studied. It is shown that under some natural conditions the solution of the interpolation problem exists and is unique. Furthermore, it is given in a simple closed form which makes it attractive for practical applications. Asymptotic analysis confirms the expected approximation order, i.e., order six. Numerical examples pave the way for a promising nonlinear geometric subdivision scheme.     

A subdivision algorithm for generalized Bernstein–Bézier curves

In this article we present a computationally efficient subdivision algorithm for the evaluation of generalized Bernstein–Bézier curves. As particular cases we have subdivision algorithms for classical as well as trigonometric Bernstein–Bézier curves.     

A generalized unconstrained theory and isogeometric finite element analysis based on Bézier extraction for laminated composite plates

This work presents an isogeometric finite element formulation based on Bézier extraction of the non-uniform rational B-splines (NURBS) in combination with a generalized unconstrained higher-order shear deformation theory (UHSDT) for laminated composite plates. The proposed approach relaxes zero-shear stresses at the top and bottom surfaces of the plates and no shear correction factors are required. A weak form of static, free vibration and transient response analyses for laminated composite plates is then established and is numerically solved using isogeometric Bézier finite elements. NURBS can be written in terms of Bernstein polynomials and the Bézier extraction operator. IGA is implemented with the presence of C°-continuous Bézier elements which allow to easily incorporate into existing finite element codes without adding many changes as the former IGA. As a result, all computations can be performed based on the basis functions defined previously as the same way in finite element method (FEM). Numerical results performed over static, vibration and transient analysis show high efficiency of the present method.     

Thermal modelling of three-way mixing valves using Bézier curves for parameter estimation applications

Each mixing valve is characterized by its k_vs-value and the valve characteristics. In mounted condition, the valve characteristics are deformed and the degree of the deformation is determined by the valve authority. Considering already installed mixing valves, neither the valve authority nor the course of the undeformed valve characteristics are exactly known. The lack of information poses a challenge when physically modelling the relationship between the inlet temperatures, the control signal and the outlet temperature.Based on Bézier curves, a universal non-linear grey-box model to describe the thermal valve behaviour is proposed. It is compared with a state space as well as a Hammerstein model and shows the advantages over black-box approaches. The trained model is very robust, able to predict measurement data very accurately and can be used in control design. The parameter estimation is performed using the Open Source platform JModelica.org.