Designing objects with a spline

Abstract

Splines play an important role in approximation theory as well as in computer-aided design and geometric modelling. Specifically, designing of objects is a fundamental problem that is mainly being attempted to be resolved using splines. Many authors have worked in the area and various techniques exist in the current literature. B-splines and non-uniform B-splines (NURBS) are quite popular in literature. However, there also exist some other splines to compete. The paper is devoted towards a new spline method that is a generalization of the Ball's cubic spline method on one hand, and also serves as an effective alternate to the NURBS of degree three on the other hand. Together with a class of shape parameters in its description, it introduces a class of cubic polynomial blending functions. These blending functions play a role of basis functions that ultimately describe the newly proposed spline method. Piecewise polynomial curves with a family of shape parameters are constructed from these blending functions. The generated curves have a second-order geometric continuity similar to those of weighted Nu-splines. The proposed method also provides a strong alternate to the NURBS of degree three as far as shape control is concerned. If the values of the shape parameters are changed, the approaching degree of the curves to their control polygon is adjusted accordingly and the curves are manipulated to approximate the cubic B-spline curve from its both sides. In comparison with the existing techniques, the degree of the spline method is ideally three and the domain of the shape parameters is broader than those in NURBS. The scheme has been implemented practically to demonstrate its effectiveness.This curve design method, in addition to enjoying the good features of Ball's cubic splines, possesses interesting shape design features too. The proposed spline method possesses two families of shape parameters. One family of parameters is associated with intervals and the other with points to have variety of shape control.

Keywords:

• spline
• approximation
• computer-aided design
• blending functions
• geometric continuity

2000 AMS Subject Classification :

• 68U05
• 68U07

Acknowledgements

This work has been supported by Kuwait University.