Control Points Of Curve Research Articles

B-Spline Curve Fitting Based on Adaptive Particle Swarm Optimization Algorithm

For fitting of ordered plane data by B-spline curve with the least squares, the genetic algorithm is generally used, accompanying the optimization on both the data parameter values and the knots to result in good robust, but easy to fall into local optimum, and without improved fitting precision by increasing the control points of the curve. So what we have done are: combining the particle swarm optimization algorithm into the B-spline curve fitting, taking full advantage of the distribution characteristic for the data, associating the data parameters with the knots, coding simultaneously the ordered data parameter and the number of the control points of the B-spline curve, proposing a new fitness function, dynamically adjusting the number of the control points for the B-spline curve. Experiments show the proposed particle swarm optimization method is able to adaptively reach the optimum curve much faster with much better accuracy accompanied less control points and less evolution times than the genetic algorithm.

Fast algorithm for surface reconstruction from cloud data based RBF neural network

On the basis of the analysis of the existing reconstruction methods' limitations,a fast neural network based algorithm for 3D surface reconstruction from cloud data was presented.Firstly,a unitary processing for the cloud data was made.And then the contour lines were extracted and the surface based contour lines were segmented.The method can directly from the neural network value matrix get the right curve control points or surface control mesh,and through neural network values restriction achieve the right curve or patch of smooth connection.Experimental results show that this method can quickly obtain good shape Mesh.

Bézier curves and [formula omitted] interpolation in Riemannian manifolds

In a connected Riemannian manifold, generalised Bézier curves are C ∞ curves defined by a generalisation, in which line segments are replaced by minimal geodesics, of the classical de Casteljau algorithm. As in Euclidean space, these curves join their first and last control points. We compute the endpoint velocities and (covariant) accelerations of a generalised Bézier curve of arbitrary degree and use the formulae to express the curve's control points in terms of these quantities. These results allow generalised Bézier curves to be pieced together into C 2 splines, and thereby allow C 2 interpolation of a sequence of data points. For the case of uniform splines in symmetric spaces, we show that C 2 continuity is equivalent to a simple relationship, involving the global symmetries at knot points, between the control points of neighbouring curve segments. We also present some examples in hyperbolic 2-space.

Shape modification of Bézier curves by constrained optimization

The Bezier curve is one of the most commonly used parametric curves in CAGD and Computer Graphics and has many good properties for shape design. Developing more convenient techniques for designing and modifying Bezier curve is an important problem, and is also an important research issue in CAD/CAM and NC technology fields. This work investigates the optimal shape modification of Bezier curves by geometric constraints. This paper presents a new method by constrained optimization based on changing the control points of the curves. By this method, the authors modify control points of the original Bezier curves to satisfy the given constraints and modify the shape of the curves optimally. Practical examples are also given.

Cross-sectional design with curvature constraints

A practical example of B-spline curve control points manipulation for the geometric construction of a free form shape is presented. Elements of a cross-sectional design methodology are used in conjunction with a skinning type operator for the definition of a B-spline surface. Skinning process is well established in the CAD community, but further difficulties arise in producing smooth surfaces under constraints. This paper attempts to overcome the fairness problem by choosing an appropriate solution where the execution time has to be reasonably short. Main results include an industrial application in a preliminary aerodynamic design cycle where manufacturing tolerances defined by smoothness criteria are maintained.

Fast Detection of the Geometric Form of Two-Dimensional Cubic Bézier Curves

Abstract This paper presents an algorithm that determines, from the control points of a cubic two-dimensional Bezier curve, the geometric form of the curve (arch, single inflecti on, two inflections, cusp, loop, or the degenerate forms straight line or point). The algorithm is based on a case analysis of the signs of the determinants of the control-point triples. Source code is available on the web site listed at the end of this paper.

Recursive algebraic curve fitting and rendering

We describe a recursive algorithm that uses quadratic algebraic curve segments to vectorize digital images. The closeness of fitting and the smoothness of connection between curve segments are ensured by a recursive algebraic curve fitting and a subsequent fine-tuning procedure. The idea is to provide an alternative way to vectorize outside parametric schemes, while maintaining the precision of parametric vectorization. We can also have all the new features of algebraic representation; for instance, the implicit forms and unique insights into curve shapes and control point weights. We also present a triangular quadtree rendering scheme for displaying algebraic curves. These algorithms combine features from both parametric and algebraic schemes to meet different requirements for curve fitting.

Bounds on the Moving Control Points of Hybrid Curves

Hybrid curves provide an attractive method for approximating rational Bézier curves by polynomial Bézier curves. In this paper, several methods are provided to estimate the error bounds for the approximation to the moving control point of the hybrid curves. When the given rational Bézier curves satisfies the convergent conditions for moving control point of the hybrid curve, by these methods we can choose a hybrid curve with a certain degree such that the distance between the moving control point and a special point is less than a given error bound ϵ. So the polynomial Bézier curves to approximate the rational Bézier curve can be obtained by replacing the moving control point with the special point.

Fuzzy B-spline membership function (BMF) and its applications in fuzzy-neural control

A general methodology for constructing fuzzy membership functions via B-spline curves is proposed. By using the method of least-squares, the authors translate the empirical data into the form of the control points of B-spline curves to construct fuzzy membership functions. This unified form of fuzzy membership functions is called a B-spline membership function (BMF). By using the local control property of a B-spline curve, the BMFs can be tuned locally during the learning process. For the control of a model car through fuzzy-neural networks, it is shown that the local tuning of BMFs can indeed reduce the number of iterations tremendously. This fuzzy-neural control of a model car is presented to illustrate the performance and applicability of the proposed method. >

Collision-Free Trajectory Generation for Robot Manipulator by Optimizing B-Spline Control Points.

A new algorithm is proposed to generate a collision-free trajectory for a robot manipulator. In this method, joint trajectories are described by uniform B-spline curves and the control points which represent initially specified joint trajectories are improved by the nonlinear optimization technique called Complex Method. The algorithm makes use of two advantages. The first one is that control points of B-spline curves uniquely, continuously, and locally describe joint trajectories. The second one is that the Complex Method does not require the gradient of performance index which represents the interference between manipulator links and obstacles and which evaluates the goodness of the trajectories. The proposed algorithm is simple and easily applicable. The effectiveness is also confirmed by examples for the trajectory generation of multijoint manipulators.

Using general polar values as control points for polynomial curves

Blossoming has proven to be a useful technique for understanding and generalizing polynomial curves through the use of the polar form. This paper shows that general polar values may be used to control polynomial curves when a related matrix is invertible. The inverse matrix provides a useful translation from these general blossom control points to well known ones such as those of Bézier. The special case in which the polar form can be evaluated through pairwise affine combinations is characterized, allowing the arguments of the blossom control points to be chosen in a manner akin to choosing the knot vector of a B-spline segment. The number of free parameters for specifying the blossom control points of polynomial curves is increased significantly over the B-spline case.

Joining rational curves smoothly

A theory is developed for projective-invariant construction from a sequenceof control points of a piecewise rational curve of arbitrary degree joining with continuity of certain geometric properties of the curve. In particular a recursive means of evaluation is derived which generalises the Cox-de Boor algorithm for B-splines.

Bézier curves of positive curvature

Sufficient conditions on the control points of an nth degree curve are given which insure that the curve has positive curvature. The conditions allow greater a priori control of the shape of parametric curves.