A multivariate Powell-Sabin interpolant

We consider the problem of constructing a C1 piecewise quadratic interpolant, Q, to positional and gradient data defined at the vertices of a tessellation of n-simplices in IR. The key to the interpolation scheme is to appropriately subdivide each simplex to ensure that certain necessary geometric constraints are satisfied by the subdivision points. We establish these constraints using the Bernstein–Bézier form for polynomials defined over simplices, and show how they can be satisfied. When constructed, the interpolant Q has full approximation power.