Abstract. We introduce the embedded Weingarten map as a geometric invariant of piecewise smooth surfaces. It is given by a (3 × 3)-matrix and provides complete curvature information in a continuous way. Thus, it is the appropriate tool for the C 2-analysis of subdivision surfaces near extraordinary points. We derive asymptotic expansions and show that the convergence of the sequence of embedded Weingarten maps to a constant limit is necessary and sufficient for curvature continuity. 