Constructing cartesian splines

TL;DR

This paper introduces Cartesian splines (C-splines), a new class of piecewise polynomials defined on Cartesian coordinate systems that ensure a specified level of continuity by constraining polynomial coefficients through a null-space approach.

Contribution

It presents the derivation of C-spline bases and an algorithm for constructing C-spline models, enabling automatic enforcement of continuity across polynomial segments.

Findings

C-splines are piecewise polynomials with enforced Cr continuity.

The null-space approach simplifies the construction of continuous spline models.

An algorithm for efficient C-spline model construction is provided.

Abstract

We introduce here Cartesian splines or, for short, C-splines. C- splines are piecewise polynomials which are defined on adjacent Cartesian coordinate systems and are Cr continuous throughout. The Cr continuity is enforced by constraining the coefficients of the polynomial to lie in the null-space of some smoothness matrix H. The matrix-product of the null-space of the smoothness matrix H and the original polynomial base results in a new base, the so-called C-spline base, which automatically enforces Cr continuity throughout. In this article we give a derivation of this C-spline base as well as an algorithm to construct C-spline models.