We construct a suitable B-spline representation for a family of bivariate spline functions with smoothness r and polynomial degree 3r-1. They are defined on a triangulation with Powell-Sabin refinement. The basis functions have a local support, they are nonnegative and they form a partition of unity. The construction involves the determination of triangles that must contain a specific set of points. We further consider a number of CAGD applications. We show how to define control points and tangent control polynomials (of degree 2r-1), and we provide an efficient and stable computation of the Bernstein-Bézier form of such splines.