Abstract: In this paper, a unified model for approximating smooth or rough surfaces defined in the triangular field is introduced. This model is called triangular projected iterated function system (TPIFS). The model combines the classical iterated function system (IFS) model and the free form approximation theory, and it is also proved that the model can approximate any continuous functions defined in the triangular field. The main applications of the model are surface modeling, shape description and image compression.