In this paper, parametric polynomial minimal surfaces of degree six with isothermal parameter are discussed. We firstly propose the sufficient and necessary condition of a harmonic polynomial parametric surface of degree six being a minimal surface. Then we obtain two kinds of new minimal surfaces from the condition. The new minimal surfaces have similar properties as Enneper’s minimal surface,...

A generalization of Patterson’s work (Patterson, 1985), on the invariants of the rational Bézier curves, to the case of surfaces is presented. An equation for the determination of the invariants for surfaces of degree (n, n) is derived and solved for the bi-quadratics – for which it is shown that seven independent, invariant functions exist. Explicit forms of the invariants are derived and a nu...

The Plateau–Bézier problem consists in finding the Bézier surface with minimal area from among all Bézier surfaces with prescribed border. An approximation to the solution of the Plateau–Bézier problem is obtained by replacing the area functional with the Dirichlet functional. Some comparisons between Dirichlet extremals and Bézier surfaces obtained by the use of masks related with minimal surf...

We discuss 3D Bézier curves with monotone curvature and torsion, generalizing a 2D class of curves by Y. Mineur et al. © 2006 Elsevier B.V. All rights reserved.

It is well known that the degree-raised Bernstein-Bézier coefficients 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 coefficients. We show how An can be viewed as an inverse to the Bernstein operator and that the derivatives An(g) (r) converge uniformly to g at the rate 1/n for all r. We also give an as...

We provide control point based necessary and sufficient conditions for (n,m) Bézier surfaces to have linear isoparametric lines. © 2007 Elsevier B.V. All rights reserved.

In this paper we present a G2 end-point interpolation of offset curves using rational Bézier curves of degree seven. The method is based on a G2 end-point interpolation of circular arcs using biquadratic Bézier curves. We also prove the invariance of the Hausdorff distance between two compatible curves under convolution. Using this result, we obtain the exact Hausdorff distance between an offse...

This paper proposes a compensated algorithm to evaluate Bézier tensor product surfaces with floating-point coefficients and coordinates. This algorithm is based on the application of error-free transformations to improve the traditional de Casteljau tensor product algorithm. This compensated algorithm extends the compensated de Casteljau algorithm for the evaluation of a Bézier curve to the cas...

Non-self-intersection is both a topological and a geometric property. Recent theorems show how non-self-intersecting Bézier curves also have non-self-intersecting control polygons, after sufficiently many uniform subdivisions. As a partial generalization, a sufficient condition is given within R for a non-self-intersecting, C cubic Bézier curve to be ambient isotopic to its control polygon form...