Conic section is one of the geometric elements most commonly used for shape expression and mechanical accessory cartography. A rational quadratic Bézier curve is just a conic section. It cannot represent an elliptic segment whose center angle is not less than . However, conics represented in rational quartic format when compared to rational quadratic format, enjoy better properties such as bein...

In order to eliminate the weakness of large volumes data in 3D modeling process, a fast and efficient sampling method based on curvature analysis of cubic Bézier curve is presented. With the method, the characteristic points and curvature maximum value of cubic Bézier curve are used as the reference standard of the sampling interval and density. For cubic B-spline curves, the sampling data can ...

The preservation of ambient isotopic equivalence under piecewise linear (PL) approximation for smooth knots are prominent in molecular modeling and simulation. Sufficient conditions are given regarding: (1) Hausdorff distance, and (2) a sum of total curvature and derivative. High degree Bézier curves are often used as smooth representations, where computational efficiency is a practical concern...

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