Farin points (weight points) are a useful tool for handling the weights of rational Bézier curves. They describe the weights of the Bézier points uniquely and in a geometrically intuitive way. The main problem for using Farin points for triangular or tensorproduct rational Bézier surfaces is the fact that they are not independent of each other and therefore overdefine the weights. To overcome t...

In the present paper we investigate rational two-parameter families of spheres and their envelope surfaces in Euclidean R3. The four dimensional cyclographic model of the set of spheres in R3 is an appropriate framework to show that a quadratic triangular Bézier patch in R4 corresponds to a two-parameter family of spheres with rational envelope surface. The construction shows also that the enve...

A tensor-product Bézier surface patch x of degree (m,n) is called biharmonic if it satisfies 2x = 0. As shown by Monterde and Ugail (2004), these surface patches are fully determined by their four boundaries. In this note we derive necessary conditions for their existence. © 2006 Elsevier B.V. All rights reserved.

Properties of a parametric curve in R 3 are often determined by analysis of its piecewise linear (PL) approximation. For Bézier curves, there are standard algorithms, known as subdivision, that recursively create PL curves that converge to the curve in distance. The exterior angles of PL curves under subdivision are shown to converge to 0 at the rate of O(1 2 i), where i is the number of subdiv...

As a new method of representing curves, Q-Bézier curves not only exhibit the beneficial properties of Bézier curves but also allow effective shape adjustment by changing multiple shape parameters. In order to resolve the problem of not being able to construct complex curves using a single curve, we study the geometric continuity conditions for Q-Bézier curves of degree n. Following the analysis...

Presented paper describes the method for finding the intersection of class space rational Bezier curves. The problem curve/curve intersection belongs among basic geometric problems and the aim of this article is to describe the new technique to solve the problem using relief perspective and Bezier clipping.

The paper is devoted to the parametrization extension problem: given a loop composed of 3 (resp. 4) rational Bézier curves on a rational surface X, find a triangular (resp. tensor product) Bézier patch on X of optimal degree bounded by these curves. The constructive solution to the formulated problem is presented in details for cases where X is a sphere or a hyperbolic paraboloid. Then X is an ...

Two bounds on the magnitude of the derivative of rational Bézier curves are presented and compared with the known ones. As an application, we improve the existing bounds on the magnitude of partial derivatives of rational Bézier surfaces.  2005 Elsevier B.V. All rights reserved.

In this talk we show a construction for characterising developable surfaces in the form of Bézier triangular patches. It is shown that constructions used for rectangular patches are not useful, since they provide degenerate triangular patches. Explicit constructions of nondegenerate developable triangular patches are provided.