A piecewise quintic G1 spline surface interpolating the vertices of a triangular surface mesh of arbitrary topological type is presented. The surface has an explicit triangular Bezier representation, is affine invariant and has local support. The twist compatibility problem which arises when joining an even number of polynomial patches G1 continuously around a common vertex is solved by constructing C2-consistent boundary curves. Piecewise C1 boundary curves and a regular 4-split of the domain triangle make shape parameters available for controlling locally the boundary curves. A small number of free inner control points can be chosen for some additional local shape effects.