The goal of this paper is to derive linear convexity conditions for Bernstein–Bézier surfaces defined on rectangles and triangles. Previously known linear conditions are improved on, in the sense that the new conditions are weaker. Geometric interpretations are provided.

We analyze the class of surfaces which are equipped with rational support functions. Any rational support function can be decomposed into a symmetric (even) and an antisymmetric (odd) part. We analyze certain geometric properties of surfaces with odd and even rational support functions. In particular it is shown that odd rational support functions correspond to those rational surfaces which can...

This note derives formulae for Gaussian and mean curvatures for tensor-product and triangular rational Bézier patches in terms of the respective control meshes. These formulae provide more geometric intuition than the generic formulae from differential geometry.

Given a prescribed boundary of a Bézier surface, we compare the Bézier surfaces generated by two different methods, i.e., the Bézier surface minimising the biharmonic functional and the unique Bézier surface solution of the biharmonic equation with prescribed boundary. Although often the two types of surfaces look visually the same, we show that they are indeed different. In this paper, we prov...

In this work, we develop a framework based on piecewise Bézier curves to plane shapes deformation and we apply it to shape optimization problems. We describe a general setting and some general result to reduce the study of a shape optimization problem to a finite dimensional problem of integration of a special type of vector field. We show a practical problem where this approach leads to effici...

Article history: Received 20 February 2008 Received in revised form 21 February 2011 Accepted 21 March 2011 Available online 23 March 2011

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...