De Casteljau Algorithm Research Articles

The De Casteljau Algorithm on Lie Groups and Spheres

We examine the De Casteljau algorithm in the context of Riemannian symmetric spaces. This algorithm, whose classical form is used to generate interpolating polynomials in {\Bbb R}^n, was also generalized to arbitrary Riemannian manifolds by others. However, the implementation of the generalized algorithm is difficult since detailed structure, such as boundary value expressions, has not been available. Lie groups are the most simple symmetric spaces, and for these spaces we develop expressions for the first and second order derivatives of curves of arbitrary order obtained from the algorithm. As an application of this theory we consider the problem of implementing the generalized De Casteljau algorithm on an m-dimensional sphere. We are able to fully develop the algorithm for cubic splines with Hermite boundary conditions and more general boundary conditions for arbitrary m.

Error analysis of corner cutting algorithms

Corner cutting algorithms are used in different fields and, in particular, play a relevant role in Computer Aided Geometric Design. Evaluation algorithms such as the de Casteljau algorithm for polynomials and the de Boor–Cox algorithm for B‐splines are examples of corner cutting algorithms. Here backward and forward error analysis of corner cutting algorithms are performed. The running error is also analyzed and as a consequence the general algorithm is modified to include the computation of an error bound.

Nonlinear corner‐cutting

This begins the study of a Riemannian generalization of a special case of algorithm I of Lane and Riesenfeld (1980), closely related to the de Casteljau algorithm (Goldman, 1989) for generating cubic polynomial curves. In our version, as in Shoemake's (1985), straight lines are replaced by geodesic segments. Our construction differs from Shoemake's in that it is a kind of stationary subdivision algorithm, defined by a recursive procedure, and it is not at all clear from the construction that a limiting curve q∞ exists, much less that it is differentiable. Indeed, the aim of the present paper is to prove that q∞ is differentiable and that the derivative is Lipschitz. The result is nontrivial: it is well‐known that stationary subdivision typically defines non‐differentiable curves (Cavaretta et al., 1991). On the other hand Shoemake's algorithm is non‐recursive and evidently defines a C∞ curve. Other approaches to splines on curved spaces are considered in (Barr et al., 1992; Chapman and Noakes, 1991; Duff, 1985; Gabriel and Kajiya, 1985; Noakes et al., 1989).

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.

A de Casteljau algorithm for generalized Bernstein polynomials

This paper is concerned with a generalization of the classical Bernstein polynomials where the function is evaluated at intervals which are in geometric progression. It is shown that these polynomials can be generated by a de Casteljau algorithm, which is a generalization of that relating to the classical case.

Bernstein-Be´zier Harmonics and Their Application to Robot Trajectory Synthesis With Minimal Vibrational Excitation

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. This paper presents a Bernstein-Be´zier form of harmonic trajectory patterns for synthesizing low-harmonic trajectories. Similar to Bernstein-Be´zier polynomial curves, Bernstein-Be´zier harmonic trajectories can be defined either explicitly using Bernstein-Be´zier basis harmonics or recursively using the harmonic deCasteljau algorithm. The second part of the paper demonstrates how a Bernstein-Be´zier trajectory can be combined with the inverse dynamics of a robot manipulator for synthesizing a joint trajectory that demands “minimal” dynamic response from its actuators. An example involving a planar 2R robot is also presented.

Recursive calculation of the derivatives of rational curves in (BR) form

Several recurrence formulae are presented to calculate the derivatives of any order of a rational curve in the ( BR ) form in terms of its control polygon. This method is based on an extension to the ( BR ) curves of the De Casteljau algorithm.

Properties of two types of generalized ball curves

Ball basis was introduced for cubic polynomials by Ball, and was generalized for polynomials of higher degree by Wang and Said. In this paper, we investigate degree elevation and reduction, recursive algorithm, enveloping theorem, and Bézier representation of these two generalized Ball curves. Furthermore, the efficiency of de Casteljau algorithm for Bézier curves and recursive algorithms for two generalized Ball curves is compared.

De Casteljau's algorithm is an extrapolation method

One of the most important recursive schemes in CAGD is De Casteljau's algorithm for the evaluation of Bézier curves and surfaces. Within the theory of triangular recursive schemes we discuss the De Casteljau's algorithm as a particular case, i.e. we prove that it is identical to the E-algorithm (or GNA-algorithm) in a particular frame. This result is of theoretical interest since it leads to some classification of recurrence relations in CAGD. Furthermore, it may be regarded as a model example how to obtain known and possibly new recursive schemes in CAGD as examples of the theory of general extrapolation algorithms.

Geometric Construction of Be´zier Motions

This paper deals with discrete computational geometry of motion. It combines concepts from the fields of kinematics and computer aided geometric design and develops a computational geometric framework for geometric construction of motions useful in mechanical systems animation, robot trajectory planning and key framing in computer graphics. In particular, screw motion interpolants are used in conjunction with deCasteljau-type methods to construct Be´zier motions. The properties of the resulting Be´zier motions are studied and it is shown that the Be´zier motions obtained by application of the deCasteljau construction are not, in general, of polynomial type and do not possess the useful subdivision property of Bernstein-Be´zier curves. An alternative from of deCasteljau algorithm is presented that results in Be´zier motions with subdivision property and Bernstein basis function. The results are illustrated by examples.

An affine representation of de Casteljau's and de Boor's rational algorithms

The well-known algorithm of de Casteljau was already known in 1870 by I.C.F. Haase. Haase's idea is strongly related to an affine representation of de Casteljau's algorithm for rational curves and surfaces, and even to de Boor's algorithm for rational splines.

Efficient Algorithms for Periodic Hermite Spline Interpolation

Periodic Hermite spline interpolants on an equidistant lattice are represented by the Bézier technique as well as by the B-spline method. Circulant matrices are used to derive new explicit formulas for the periodic Hermite splines of degree m and defect $r\;(1 \leq r \leq m)$. Applying the known de Casteljau algorithm and the de Boor algorithm, respectively, we obtain new efficient real algorithms for periodic Hermite spline interpolation.

Integer de Casteljau Algorithm for Rasterizing NURBS Curves

AbstractAn integer version of the well‐known de Casteljau algorithm of NURBS curves is presented here. The algorithm is used to render NURBS curves of any degree on a raster device by turning on pixels that are closest to the curve. The approximation is independent of the parametrization, that is, it is independent of the weights used. The algorithm works entirely in the screen coordinate system and produces smooth rendering of curves without oversampling. Because of the integer arithmetic used, the algorithm is easily cast in hardware.

An efficient algorithm for subdividing linear Coons surfaces

An efficient algorithm for subdividing linear Coons surfaces with cubic Bezier curves as boundaries is described. Algorithms for subdividing bilinear surfaces and ruled surfaces are also considered since they are used to develop the algorithm. The algorithm is faster than the algorithm for subdividing a patch in Bezier form via the deCasteljau algorithm. The minmax box of the subdivided Coons surface is derived in order to perform intersection calculations by means of subdivision.

A generalized Ball curve and its recursive algorithm

The use of Bernstein polynomials as the basis functions in Bézier's UNISURF is well known. These basis functions possess the shape-preserving properties that are required in designing free form curves and surfaces. These curves and surfaces are computed efficiently using the de Casteljau Algorithm. Ball uses a similar approach in defining cubic curves and bicubic surfaces in his CONSURF program. The basis functions employed are slightly different from the Bernstein polynomials. However, they also possess the same shape-preserving properties. A generalization of these cubic basis functions of Ball, such that higher order curves and surfaces can be defined and a recursive algorithm for generating the generalized curve are presented. The algorithm could be extended to generate a generalized surface in much the same way that the de Casteljau Algorithm could be used to generate a Bézier surface.

Apex: Two architectures for generating parametric curves and surfaces

The interactive design of parametric curves and surfaces places a tremendous computational burden on general-purpose workstations. We describe two architectures for a VLSI co-processor chip that generates a large class of spline descriptions extremely quickly. This architecture is based on a generalization of the de Casteljau algorithm for Bezier curves and the de Boor algorithm forB-splines that generates points on a curve or surface in a data-flow fashion. The first chip, Apex I, maps the data-flow structure directly into silicon, allowing it to generate curves and surfaces at a rate approaching two million points per second. The second chip, Apex II, performs the same computation in a more flexible way that allows the generation of higher degree curves at the cost of lower performance. This paper only briefly reviews the theory underlying the architecture, focusing instead on the design and implementation of the Apex chips.

A multisided generalization of Bézier surfaces

In this paper we introduce a class of surface patch representations, called S-patches, that unify and generalize triangular and tensor product Bézier surfaces by allowing patches to be defined over any convex polygonal domain; hence, S-patches may have any number of boundary curves. Other properties of S-patches are geometrically meaningful control points, separate control over positions and derivatives along boundary curves, and a geometric construction algorithm based on de Casteljau's algorithm. Of special interest are the regular S-patches, that is, S-patches defined on regular domain polygons. Also presented is an algorithm for smoothly joining together these surfaces with C k continuity.

The numerical problem of using Be´zier curves and surfaces in the power basis

The computations of curves and surfaces points for CAD modeling are numerous and important. In the case of modeling using the Be´zier method, these may be achieved either with the De Casteljau algorithm in the Bernstein basis, or with the Horner algorithm in the power basis. The De Casteljau algorithm requires a greater number of operations than Horner's. However, we show that the equations of curves and surfaces in the power basis may be affected by a very important loss of significant digits on the polynomials coefficient; this is due to the required conversion matrices which are ill-conditioned. Examples are given. We conclude that the use of the Horner algorithm should be avoided for the computations of curves and surfaces points with the Be´zier method.

Uniform refinement of curves

We propose and analyze a class of algorithms for the generation of curves and surfaces. These algorithms encompass some well-known methods of subdivision for Bernstein-Bézier curves (de Casteljau's algorithm) and B-spline curves (Lane and Riesenfeld's algorithm). Several results concerning properties of the limiting curves as well as related questions are discussed.

A Top Down Method for Interactive Drawing

AbstractTraditional interactive drawing programs adopt a bottom‐up approach, allowing the user to construct a picture by the use of discrete tools, for example, lines, circles, rectangles, and so on. This paper presents a different approach, which allows users to construct graphical objects by stretching and cutting existing objects. The representation is simply implemented, based on a ring of cubic Bezier curves, and use of the de Casteljau algorithm.