An Improvement on the Upper Bounds of the Partial Derivatives of NURBS Surfaces

Ye Tian,Zhitong Chen,Jianmin Zheng,Tao Ning,Jixing Li

doi:10.3390/math8081382

Abstract

The Non-Uniform Rational B-spline (NURBS) surface not only has the characteristics of the rational Bézier surface, but also has changeable knot vectors and weights, which can express the quadric surface accurately. In this paper, we investigated new bounds of the first- and second-order partial derivatives of NURBS surfaces. A pilot study was performed using inequality theorems and degree reduction of B-spline basis functions. Theoretical analysis provides simple forms of the new bounds. Numerical examples are performed to illustrate that our method has sharper bounds than the existing ones.

Highlights

The bounds on derivatives of Non-Uniform Rational B-spline (NURBS) surfaces have very important applications for ComputerAided Geometric Design (CAGD)
Since Floater [2] gave a classical inequality to estimate the upper bounds of the first-order derivatives of Bézier curves, several improved results have been derived for rational Bézier curves and surfaces [3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18]
This paper presents a novel method for estimating the bounds of the first- and second-order partial derivatives of NURBS

Summary

Introduction

The bounds on derivatives of NURBS surfaces have very important applications for Computer. Cao [20], Hu [21] and Liu [22] presented methods for the bounds estimation of the second-order derivatives of rational triangular Bézier surfaces. Wang [23] obtained the upper bounds expression of the third-order derivatives of the rational Bézier curves by moving control points. The bounds on first-order derivatives of NURBS surfaces given in this paper are sharper than the existing ones [24,25] They are less affected by the change of weights, knot vectors and control points. In the piecewise linear approximation algorithm proposed by Filip et al [26], a more convergent derivative bound can make the result of surface subdivision more accurate Both theoretical analysis and numerical examples are provided to verify the superiority of the proposed method.

Preliminary

Bounds on the First-Order Partial Derivatives

Bounds on the Second-Order Partial Derivatives

Numeric Examples

Conclusions