The recent ability to measure quickly and inexpensively dense sets of points on physical objects has deeply influenced the way engineers represent shapes in CAD systems, animation software or in the game industry. Many researchers advocated to completely bypass smooth surface representations, and to stick to a dense mesh model throughout the design process. Yet smooth analytic representations are still required in standard CAD systems and animation software, for reasons of compactness, control, appearance and manufacturability. In this paper we present a new method for fitting a smooth adaptively refinable triangular spline surface of arbitrary topology to an arbitrary dense triangular mesh. The key ingredient in our solution is that adaptive fitting is achieved by 4-splitting triangular surface patches locally therefore no particular attention has to be paid the validity of an underlying subdivided mesh. Furthermore, the final surface is composed of low-degree polynomial patches that always join with G1-continuity. The ability to adaptively refine the model allows to achieve a given approximation error with a minimal number of patches.