Pairs of B-Splines with Small Support on the Four-Directional Mesh Generating a Partition of Unity

Let τ be the four-directional mesh of the plane and let Σ1 (respectively Λ1) be the unit square (respectively the lozenge) formed by four (respectively eight) triangles of τ. We study spaces of piecewise polynomial functions in Ck(R2) with supports Σ1 or Λ1 having sufficiently high degree n, which are invariant with respect to the group of symmetries of Σ1 or Λ1 and whose integer translates form a partition of unity. Such splines are called complete Σ1 and Λ1-splines. We first give a general study of spaces of linearly independent complete Σ1 and Λ1-splines of class Ck and degree n. Then, for any fixed k≥0, we prove the existence of complete Σ1 and Λ1-splines of class Ck and minimal degree, but they are not unique. Finally, we describe algorithms allowing to compute the Bernstein–Bézier coefficients of these splines.