Compensated Evaluation of Tensor Product Surfaces in CAGD

In computer-aided geometric design, a polynomial surface is usually represented in Bézier form. The usual form of evaluating such a surface is by using an extension of the de Casteljau algorithm. Using error-free transformations, a compensated version of this algorithm is presented, which improves the usual algorithm in terms of accuracy. A forward error analysis illustrating this fact is developed.

Modification of the Curve and the Surface Polynomial Bezier Using de Casteljau Algorithm

<p class="Abstract">Research carried out to obtain a Bezier curve of degree six resulting curvature of the curve is more varied and multifaceted. Stages in formulating applications Bezier surfaces revolution in design, there are three marble objects. First, calculate the parametric representation revolution Bezier surface and shape modification in a number of different forms. Second, formulate Bezier parametric surfaces that are continuously incorporated. Lastly, apply the formula to the design objects using computer simulation. Results marble obtained are Bezier curves of degree six modified version of the Bezier curve of degree five and some form of revolution Bezier surfaces are varied and multifaceted.</p>

Free-form additive motions using conformal geometric algebra

Free-form motions in B-spline form can be created from a number of prescribed control poses using the de Casteljau algorithm. With poses defined using conformal geometric algebra, it is natural to combine poses multiplicatively. Additive combinations offer alternative freedoms in design and avoid dealing with noninteger exponents. This paper investigates additive combinations and shows how to modify the conventional conformal geometric algebra definitions to allow such combinations to be well-defined. The additive and multiplicative approaches are compared and in general they generate similar motions, with the additive approach offering computational simplicity.

L-SYSTEMS IN GEOMETRIC MODELING

We show that parametric context-sensitive L -systems with affine geometry interpretation provide a succinct description of some of the most fundamental algorithms of geometric modeling of curves. Examples include the Lane-Riesenfeld algorithm for generating B -splines, the de Casteljau algorithm for generating Bézier curves, and their extensions to rational curves. Our results generalize the previously reported geometric-modeling applications of L -systems, which were limited to subdivision curves.

A formal study of Bernstein coefficients and polynomials

Bernstein coefficients provide a discrete approximation of the behaviour of a polynomial inside an interval. This can be used, for example, to isolate the real roots of polynomials. We prove formally a criterion for the existence of a single root in an interval and the correctness of the de Casteljau algorithm for computing Bernstein coefficients efficiently.

A de Casteljau Algorithm for -Bernstein-Stancu Polynomials

This paper is concerned with a generalization of the -Bernstein polynomials and Stancu operators, 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 and -Bernstein case.