Abstract: We present a sufficient criterion for the Bernstein Bézier (BB) form of a trivariate polynomial within a tetrahedron, such that the real zero contour of the polynomial defines a smoothand single-sheeted algebraic surface patch, We call this an A-patch. We present algorithms to build a mesh of cubic A-patches to interpolate a given set of scattered point data in three dimensions, respecting tbe topology of any surface triangulation T of the given point set. In these algorithms we first specify "normals" an the data points, then build a simplicial hull consisting of tetrahedral surrounding the surface triangulation T, and finally construct cubic A-patches within each tetrahedron. The resulting surface constructed is C¹ (tangent plane) continuous and single sheeted in each of the tetrahedral. We also show how to adjust the free parameters of the A-patches to achieve both local and global shape control.