Constructing cartesian splines

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.