De Casteljau Algorithm Research Articles

INTERPOLATION PROBLEMS ON RIEMANNIAN MANIFOLDS - A GEOMETRIC APPROACH -

The paper presents a geometric algorithm for the generation of smooth interpolating splines on Riemannian manifolds and extends previous work of the authors. Contrary to the variational approach to splines on manifolds, where the curves appear as solutions of highly nonlinear differential equations and geometric integration methods are required, our proposal generates explicit formulas. For low dimensional manifolds, the algorithm is visually very appealing because it resembles a simple geometric construction, somehow similar to the De Casteljau algorithm for the generation of Bezier curves. Our approach is however rather more flexible than the De Casteljau algorithm and also computationally less intensive, both on Euclidean spaces and on other Riemannian manifolds.

Integration of Jacobi and Weighted Bernstein Polynomials Using Bases Transformations

AbstractThis paper presents methods to compute integrals of the Jacobi polynomials by the representation in terms of the Bernstein — B´ezier basis. We do this because the integration of the Bernstein — B´ezier form simply corresponds to applying the de Casteljau algorithm in an easy way. Formulas for the definite integral of the weighted Bernstein polynomials are also presented. Bases transformations are used. In this paper, the methods of integration enable us to gain from the properties of the Jacobi and Bernstein bases.

Bound Estimations on Lower Derivatives of Rational Triangular Bézier Surfaces

Based on the de Casteljau algorithm for triangular patches,also using some existing identities and elementary inequalities,this paper presents two kinds of new magnitude upper bounds on the lower derivatives of rational triangular Bezier surfaces.The first one,which is obtained by exploiting the diameter of the convex hull of the control net,is always stronger than the known one in case of the first derivative.For the second derivative,the first kind is an improvement on the existing one when the ratio of the maximum weight to the minimum weight is greater than 2;the second kind is characterized as being represented by the maximum distance of adjacent control points.

Evaluation algorithms for multivariate polynomials in Bernstein–Bézier form

The evaluation of multivariate polynomials of n variables in Bernstein–Bézier form is considered. A forward error analysis for the corresponding de Casteljau algorithm and the VS algorithm is performed. We also include algorithms that simultaneously evaluate the polynomial and provide “a posteriori” error bounds, without increasing significantly the computational cost. The sharpness of our running error bounds is shown in the case of trivariate polynomials.

C2 spherical Bézier splines

The classical de Casteljau algorithm for constructing Bezier curves can be generalised to a sphere of arbitrary dimension by replacing line segments with shortest great circle arcs. The resulting s...

Formula omitted] spherical Bézier splines

The classical de Casteljau algorithm for constructing Bézier curves can be generalised to a sphere of arbitrary dimension by replacing line segments with shortest great circle arcs. The resulting spherical Bézier curves are C ∞ and interpolate the endpoints of their control polygons. In the present paper, we address the problem of piecing these curves together into C 2 splines. For this purpose, we compute the endpoint velocities and accelerations of a spherical Bézier curve of arbitrary degree and use the formulae to define control points that give the curve a desired initial velocity and acceleration. In addition, for uniform splines we establish a simple relationship between the control points of neighbouring curve segments that is necessary and sufficient for C 2 continuity. As illustration, we solve an interpolation problem involving sparse data using both the present method and a normalised polynomial interpolant. The normalised spline exhibits large variations in speed and magnitude of acceleration, whilst the spherical Bézier spline is far better behaved. These considerations are important in applications where velocities and accelerations need to moderated or estimated, notably computer animation and rigid body trajectory planning, where interpolation in the 3-sphere is a fundamental task.

Finding the point on Bezier curves with the normal vector passing an external point

Two algorithms for finding the point on non-rational/rational Bezier curves of which the normal vector passes through a given external point are presented. The algorithms are based on Bezier curves generation algorithms of de Casteljau's algorithm for non-rational Bezier curve or Farin's recursion for rational Bezier curve, respectively. Orthogonal projections from the external point are used to guide the directional search used in the proposed iterative algorithms. Using Lyapunov's method, it is shown that each algorithm is able to converge to a local minimum for each case of non-rational/rational Bezier curves. It is also shown that on convergence the distance between the point on curves to the external point reaches a local minimum for both approaches. Illustrative examples are included to demonstrate the effectiveness of the proposed approaches.

Running Error Analysis of Evaluation Algorithms for Bivariate Polynomials in Barycentric Bernstein Form

Running error analysis for the bivariate de Casteljau algorithm and the VS algorithm is performed. Theoretical results joint with numerical experiments show the better stability properties of the de Casteljau algorithm for the evaluation of bivariate polynomials defined on a triangle in spite of the lower complexity of the VS algorithm. The sharpness of our running error bounds is shown.

Spline Functions in Evaluation of Explosion Limit Curves for Gas Mixtures

The paper discusses a new method to approximate the sometimes missing apex point of the explosion limit curves of flammable substances with diluents in air. The base of the new method is to vary the frame points of the co-monotonic splines using de Casteljau algorithm. We show several examples for flammable/inert/oxidising gas containing systems – selected by the program TRIANGLE – where the method was applied. Due to the definition of the frame of splines it can be stated that the new method never restricts the explosion range around the apex and shifts the explosion limit curve into the direction of higher inert gas concentrations. This means that the new method can correct the highly “cut down nose” of the co-monotonic splines and gives a safer explosion range of these systems.

Rational Be´zier Line-Symmetric Motions

This paper brings together line geometry, kinematic geometry of line-symmetric motions, and computer aided geometric design to develop a method for geometric design of rational Be´zier line-symmetric motions. By taking advantage of the kinematic geometry of a line-symmetric motion, the problem of synthesizing a rational Be´zier line-symmetric motion is reduced to that of designing a rational Be´zier ruled surface. In this way, a recently developed de Casteljau algorithm for line-geometric design of ruled surfaces can be applied. An example is presented in which the Bennet motion is represented as a rational Be´zier line-symmetric motion whose basic surface is a hyperboloid.

A Note on the Connection Between Bezier Curves and Linear Optimal Control

We show that Bezier curves can in fact be thought of as the solutions to linear optimal control problems, using results from Hermite interpolation in combination with traditional linear optimal control. This provides us with a computational view of Bezier curves that differs from the standard DeCasteljau Algorithm, and it furthermore points out the close relationship between Bezier curves and interpolating dynamic splines.

Geometric Motion Control for a Kinematically Redundant Robotic Chain: Application to a Holonomic Mobile Manipulator

AbstractFor kinematically redundant robotic manipulators, the extra degrees of freedom available allows freedom in the generation of the trajectories of the end‐effector. In this paper, for this scope, we use techniques for motion control of rigid bodies on Riemannian manifolds (and Lie groups in particular) to design workspace control algorithms for the end‐effector of the robotic chain and then to pull them back to joint space, all respecting the different geometric structures of the two underlying model spaces. The trajectory planner makes use of geometric splines. Examples of the different kinds of curves that are obtained via the De Casteljau algorithm in correspondence of different metric structures in SE(3) are reported. The feedback module, instead, consists of a Lyapunov based PD controller defined from a suitable notion of error distance on the Lie group. The motivating application of our work is a holonomic mobile manipulator for which simulation results are described in detail. © 2003 Wiley Periodicals, Inc.

A unified rounding error bound for polynomial evaluation

We present a unified rounding error bound for polynomial evaluation. The bound presented here takes the same general form for the evaluation of a polynomial written in any polynomial basis when the evaluation algorithm can be expressed as a linear recurrence or a first-order linear matrix recurrence relation. Examples of these situations are: Horner's algorithm in the evaluation of power series, Clenshaw's and Forsythe's algorithms in the evaluation of orthogonal polynomial series, de-Casteljau's algorithm for Bernstein polynomial series, the modification of Clenshaw's algorithms in the evaluation of Szegő polynomial series, and so on.

Developable Bézier patches: properties and design

Geometric design of quadratic and cubic developable Bézier patches from two boundary curves is studied in this paper. The conditions for developability are derived geometrically from the de Casteljau algorithm and expressed as a set of equations that must be fulfilled by the Bézier control points. This set of equations allows us to infer important properties of developable Bézier patches that provide useful parameters and simplify the solution process for the patch design. With one boundary curve freely specified, five more degrees of freedom are available for a second boundary curve of the same degree. Various methods are introduced that fully utilize these five degrees of freedom for the design of general quadratic and cubic developable Bézier patches in 3D space. A more restricted generalized conical model or cylindrical model provides simple solutions for higher-order developable patches.

Singularities of rational Bézier curves

We prove that if an nth degree rational Bézier curve has a singular point, then it belongs to the two ( n−1)th degree rational Bézier curves defined in the ( n−1)th step of the de Casteljau algorithm. Moreover, both curves are tangent at the singular point. A procedure to construct Bézier curves with singularities of any order is given.

Rational Wang?Ball curves

Although rational Bézier curves have been commonly used as a base model for curve modelling, rational Wang?Ball curves have been found to be more efficient. The present study introduces rational Wang?Ball curves which are derived from non-rational Wang?Ball curves. It is shown that a rational Wang?Ball curve is a rational Bézier curve of the same degree and vice versa. The relationships between two rational curves established in this paper, allow conversion from one curve to the other. A recursive algorithm for plotting rational Bézier and Wang?Ball curves is obtained from the de Casteljau and Wang algorithm using homogeneous coordinates. An efficient algorithm for evaluating a rational Wang?Ball curve is attained and has a linear time complexity. Finally, another algorithm for rational Bézier curves which has linear time computation is also devised.

Blossoming beyond Extended Chebyshev Spaces

In a previous series of papers a theory of blossoming was developed for spaces of functions on an interval I spanned by the constant functions and functions Φ 1 , …, Φ n , where Φ ′ 1 , …, Φ ′ n span an extended Chebyshev space. This theory was then used to construct a generalisation of the Bernstein basis and the de Casteljau algorithm. Also considered were functions defined to be piecewise in such spaces, leading to generalisations of B-splines and the de Boor algorithm. Here we relax the condition that Φ ′ 1 , …, Φ ′ n span an extended Chebyshev space, while retaining all the nice properties of the earlier theory. This allows us to include a large variety of new spaces, including spaces of polynomials which have been found to be successful for tension methods for shape-preserving interpolation.

Efficient algorithms for Bézier curves

For evaluating Bézier curves, the de Casteljau algorithm is commonly used. Even though it is simple and stable, this algorithm is not efficient. Using the relationships between Bézier curves and two generalized Ball curves, this paper proposes three more efficient algorithms, of which the best has a linear time complexity.

Accelerating Accurate B-spline Free-form Deformation of Polygonal Objects

Abstract A previous paper described an algorithm for accurate B-spline free-form de formation of polygonal objects to producet riangular Bezier patches. However that algorithm computed the control points of the resulting patches using a generalized de Casteljaualgorithm, which is expensive to compute. In this short note, we describe an algorithm that instead uses polynomial interpolation; both theoretical analysis and implement ation results show that this new algorithmruns faster than the original.

Design and multiplierless realization of maximally flat FIR digital Hilbert transformers

A unified treatment of approximation and realization of type-3 finite impulse response (FIR) linear-phase Hilbert transformers is presented. A simple method based on Bernstein polynomials and half-band filters is proposed to derive the transfer function of the system, and a triangular array realization based on the de Casteljau algorithm is developed from the Bernstein form of the transfer function. It is shown that the array structure, consisting of multiplierless identical modules, can be realized hierarchically using complex and real signal processing techniques.