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In the present paper, a unified model for approximating smooth or rough surfaces defined in the triangular field is introduced. The model is called triangular projected iterated function system (TPIFS). TPIFS combines the classical iterated function system (IFS) model and the free form approximation theory. In addition, the properties of TPIFS and the multiresolution approximation algorithm for generating triangular surfaces rapidly are also presented. The main applications are surface modeling, shape description and geometric surface compression.