Quartic Beta-splines

Abstract

Quartic Beta-splines have third-degree arc-length or geometric continuity at simple knots and are determined by three β or shape parameters. We present a general explicit formula for quartic Beta-splines, and determine and illustrate the effects of varying the β parameters on the shape of a quartic Beta-spline curve. We show that quartic (and higher degree) rational Beta-splines with arc-length continuity satisfy the same continuity conditions as (nonrational) Beta-splines. We also show that the torsion continuous spline curves presented by Boehm ("Smooth Curves and Surfaces.” In Geometric Modeling: Algorithms and New Trends , G. E. Farin, Ed. SIAM, Philadelphia, Pa., 1987, pp. 175-184.) are equivalent to nonrational quartic Beta-spline curves, and determine the relationship between the shape parameters for the two types of curves. Finally, we present an algorithm for inserting a new knot and determining the refined control polygon.