Existence and Construction of H 1-Splines of Class C k on a Three Directional Mesh

Let be the uniform triangulation generated by the usual three directional mesh of the plane and let H ₁ be the regular hexagon formed by the six triangles of surrounding the origin. We study the space of piecewise polynomial functions in C ^k (R ²) with support H ₁ having a sufficiently high degree n, which are invariant with respect to the group of symmetries of H ₁ and whose sum of integer translates is constant. Such splines are called H ₁-splines. We first compute the dimension of this space in function of n and k. Then we prove the existence of a unique H ₁-spline of minimal degree for any fixed k0. Finally, we describe an algorithm computing the Bernstein–Bézier coefficients of this spline.