Conic Beta-Splines with Local Tension Control for Interactive Curve Fitting

Abstract

Polynomial Beta-splines were introduced by Barsky as an extension of polynomial B-splines with bias and tension parameters which allow more flexibility in controlling shape in curve fitting. It is possible to show that a quadratic Beta-spline segment is equivalent to a quadratic B-spline segment with suitably modified control vertices. This provides a simple method for evaluating quadratic Beta-splines using De Boor's algorithm for calculating polynomial B-splines. A representation for conic Beta-splines with one tension parameter is introduced and some properties are derived. They form a basis for an efficient algorithm for interactive curve fitting with conic Beta-splines. The results are extended further to cover the case of conic Beta-splines with varying tension where the tension parameter is an interpolating function between the tension values at each end of a segment.