The polar surface fluid model is used to derive the generalized dynamic shape equation and the interfacial rheological material functions for viscoelastic membranes and curved interfaces, taking viscous bending and torsion modes into full account. The materials modeling approach based on the polar surface fluid leads to the integration of bending and torsion dissipative modes with their elastic counterparts that appear in the dynamic shape equation and in the interfacial rheological functions. The covariant bending and torsion rates derived in this paper are shown to be related to the interfacial co-rotational derivative of the curvature tensor. The dynamic shape equation is used to analyze shape fluctuation in planar geometries, and to establish the role of bending dissipation in shape dynamics. The dynamic shape equation generalizes the static Helfrich shape equation by incorporating bending and torsion dissipation, and it generalizes the dynamic shape equation based on the Boussinesq-Scriven model by incorporating bending and torsion elasticity and dissipation. (c) 2006 Elsevier Inc. All rights reserved.