Non-uniform exponential tension splines

Abstract

We describe explicitly each stage of a numerically stable algorithm for calculating with exponential tension B-splines with non-uniform choice of tension parameters. These splines are piecewisely in the kernel of D 2 (D 2 − p 2 ), where D stands for ordinary derivative, defined on arbitrary meshes, with a different choice of the tension parameter p on each interval. The algorithm provides values of the associated B-splines and their generalized and ordinary derivatives by performing positive linear combinations of positive quantities, described as lower-order exponential tension splines. We show that nothing else but the knot insertion algorithm and good approximation of a few elementary functions is needed to achieve machine accuracy. The underlying theory is that of splines based on Chebyshev canonical systems which are not smooth enough to be ECC-systems. First, by de Boor algorithm we construct exponential tension spline of class C 1 , and then we use quasiOslo type algorithms to evaluate classical non-uniform C 2 tension exponential splines.

On a general new class of quasi Chebyshevian splines

“…As is well-known, such spline spaces S were initially introduced for their ability to ensure tension properties for interpolating splines, first for n = 3 and with the same positive coefficient ω everywhere [36] (the non-piecewise case). Soon after that, the general case was addressed, first in dimension four [33,37] (see also [3,17,34], ....) and then in higher dimensions.…”

Constructing totally positive piecewise Chebyshevian B-spline bases

“…In all real cases we also have τ Remark 7.6. Piecewise exponential spline spaces ( i.e., spaces of parametrically continuous splines with sections taken from any different kernels of linear differential operators all of the same orders, with constant coefficients and only real roots for their characteristic polynomials) were considered in [23] (see special cases in [20,21,7]). We obtained there both a sufficient and a necessary condition for such spline spaces to be good for design, but no necessary and sufficient condition, except concerning very specific cases.…”

Standard and Non-standard CAGD Tools for Isogeometric Analysis: A Tutorial

“…Given a general knot sequence (25), the interval [ξ p+1 , ξ n+1 ] plays a crucial role in the following 10 . Motivated by the considerations in Remark 2, we slightly modify our definition and we assume from now on that each B-spline of degree p is leftcontinuous at ξ n+1 for a general knot sequence (see also [8, Chapter IX]), i.e.,…”

“…In particular, corner cutting algorithms can be used for evaluation. Evaluation algorithms specially tuned for trigonometric and exponential GB-splines have been proposed by several authors, see for example [10,42] and references therein. Stable evaluation can also be obtained by means of non-stationary subdivision [24].…”

“…The Bernstein polynomials of degree p sum up to one and there are p + 1 of them, so we may deduce (10).…”