Degenerations Of Quadratic NURBS Curves

NURBS curve, as a curve fitting tool, is widely used in the geometric modeling. It preserves the advantages of both rational Bezier curves and B-spline curves. NURBS curves can be adjusted locally and accurately represent conic curves. When a weight tends to infinity, the limit of NURBS curve approaches to corresponding control point. However, while all the weights tend to infinity, the limit of NURBS curve is still unknown. Then the study on geometric meaning of limit of NURBS curve when all the weights tend to infinity is an interesting topic.In 2011, Garcia-Puente et al. proposed the degenerations theory of toric surfaces, which explains the geometric meaning of toric surfaces when all the weight tend to infinity.Toric surface include the rational Bezier curves and surfaces as the special cases. In this paper, by lifting functions,we define the regular control curves of quadratic NURBS curves. By means of knot insertion algorithm of NURBS curves, the curves are transformed into piecewise rational Bezier curves. When all the weights tend to infinity,we prove the limit of quadratic NURBS curve is its regular control curves.This paper is organized as follows.The first chapter introduces NURBS curve and toric degenerations theory.The second chapter recalls the basic theory of NURBS curve, including the definition and main properties, the knot insertion algorithm, and the geometric meaning of its weights.In the third chapter, we proposes the toric degenerations theory of quadratic NURBS curve. When the weights all tend to infinity, the limit curve of NURBS curve is the regular control curve and its geometric meaning of degenerations is explained.The fourth chapter of this paper provides some representative examples.