For arbitrary triangular control meshes, a surface algorithm based-on bivariate box-spline is developed.Bivariate 3-direction is a triangulation with the least directions. Box spline built on it is widely applied in CAGD.Its standard surface algorithm is only for normal control mesh in which every point has valence 6. Starting with bivariate 3-directional quartic box-splines, the paper proposes an algorithm for arbitrary triangular control meshes.The analysis of its properties especially continuity are presented in detail. The constructed surfaces by the algorithm are convex preserving, and they are piecewise C¹. The algorithm can be easily applied for global or local interpolation, which is indispensable in 3D surface reconstruction from scattered points.