Degenerations of NURBS curves while all of weights approaching infinity

TL;DR

This paper investigates the behavior of NURBS curves as all weights tend to infinity, establishing that they converge to a specific control structure called the regular control curve, with implications for shape deformation.

Contribution

It introduces the concept of regular control curves and proves their relation to NURBS limits when weights grow unbounded, extending understanding of shape deformation.

Findings

NURBS curves tend to their regular control curves as weights approach infinity.

The limit behavior depends on weights multiplied by specific functions tending to infinity.

Examples demonstrate the application of this property in shape deformation.

Abstract

NURBS curve is widely used in Computer Aided Design and Computer Aided Geometric Design. When a single weight approaches infinity, the limit of a NURBS curve tends to the corresponding control point. In this paper, a kind of control structure of a NURBS curve, called regular control curve, is defined. We prove that the limit of the NURBS curve is exactly its regular control curve when all of weights approach infinity, where each weight is multiplied by a certain one-parameter function tending to infinity, different for each control point. Moreover, some representative examples are presented to show this property and indicate its application for shape deformation.