A unified approach to non-polynomial B-spline curves based on a novel variant of the polar form

Abstract

We develop a general, unified theory of splines for a wide collection of spline spaces, including trigonometric splines, hyperbolic splines, and special Muntz spaces of splines by invoking a novel variant of the homogeneous polar form where we alter the diagonal property. Using this polar form, we derive de Boor type recursive algorithms for evaluation and differentiation. We also show that standard knot insertion procedures such as Boehm's algorithm and the Oslo algorithm readily extend to these general spline spaces. In addition, for these spaces we construct compactly supported B-spline basis functions with simple two term recurrences for evaluation and differentiation, and we show that these B-spline basis functions form a partition of unity, have curvilinear precision, and satisfy a dual functional property and a Marsden identity.