This paper proposes a new automatic mesh generation algorithm for 3D surface mesh generation. The algorithm is based on the metric specification approach and can generate anisotropic meshes on 3D surfaces. It is based on a new geometrical model using the interpolating subdivision surface concept. By using the subdivision surface concept, the new mesh generator can generate finite element meshes to model a wide range of surfaces which may contain sharp features such as cusp points and crease lines. When comparing with other traditional algorithms which use analytical surface patches as the underlying geometrical model, the new mesh generation scheme can be used in applications such as large deformation or crack analyses in which the domains to be gridded are not well defined or involve changing boundaries. The presentation of the work is divided into two parts. In Part 1, a detailed description of the underlying geometrical model used will be given, while in Part 2, attention will be focused on the mesh generation algorithms and the performance of the mesh generator developed.