It is well known that the degree-raised Bernstein-B ezier coeecients of degree n of a polynomial g converge to g at the rate 1=n. In this paper we consider the polynomial An(g) of degree n interpolating the coeecients. We show how An can be viewed as an inverse to the Bernstein polynomial operator and that the derivatives An(g) (r) converge uniformly to g (r) at the rate 1=n for all r. We also ...

Recently, we have introduced spaces of splines deened on trian-gulations lying on the sphere or on sphere-like surfaces. These spaces arose out of a new kind of Bernstein-B ezier theory on such surfaces. The purpose of this paper is to contribute to the development of a constructive theory for such spline spaces analogous to the well-known theory of polynomial splines on planar triangulations. ...

Recent work has shown that the triangular rational B ezier surface representation can be used to create polygonal surface patches. The key to the construction is the judicious use of zero weights in creating the surface parameterization. These zero weights introduce base points into the resulting rational parameterizations. Base points also lower the degree of the implicit representation of the...

We explore the computation of roots of polynomials via eigenvalue problems. In particular, we look at the case when the leading coeecient is relatively very small. We argue that the companion matrix algorithm (used, for instance, by the Matlab roots function) is inaccurate in this case. The accuracy problem is addressed by using matrix pencils instead. This improvement can be predicted from the...

This paper outlines a study into the exact solutions of (2+1)-dimensional Heisenberg ferromagnetic spin chain equation that is used to illustrate materials magnetic ordering by applying two recent techniques, namely, Sardar-subequation method and extended rational sine–cosine sinh–cosh methods. Abundant such as bright soliton, dark combined bright–dark singular soliton other periodic wave expre...

The constructionof quadraticC 1 surfacesfrom B-spline control points is generalized to a wider class of control meshes capable of outlining arbitrary free-form surfaces in space. Irregular meshes with non quadrilateral cells and more or fewer than four cells meeting at a point are allowed so that arbitrary free-form surfaces with or without boundary can be modeled in the same conceptual frame w...

The n-ary subdivision schemes contrast favorably with their binary analogues because they are capable to produce limit functions with the same (or higher) smoothness but smaller support. We present an algorithm to generate the 4-point n-ary non-stationary scheme for trigonometric, hyperbolic and polynomial case with the parameter for describing curves. The performance, analysis and comparison o...

This paper investigates the problem of path planning for a thin elastic plate. The underlying geometric model for the plate is provided by a B ezier representation. The geometric model is augmented by a realistic mechanical model. The latter permits the computation of the shape of the plate with respect to a set of grasping constraints by minimizing the elastic energy of the deformation. We use...

In contour machining of free{form surfaces with a ball{end cutter, the tool contacts the surface along loci that correspond to sections by a set of parallel planes, with equidistant spacing and common normal N. For a given tool radius, the greatest scallop height between successive tool paths is suppressed by choosing an orientation for the contouring planes that will maximize, over the entire ...