A family of tooth surfaces Sigma(phi) represented in parametric form is considered. The contents of the paper covers determination of envelope Sigma(2) to family Sigma(phi), singularities on Sigma(2) and sufficient conditions of existence of envelopes E-1 and E-2 to characteristics on generating and generated tooth surfaces Sigma(1) and Sigma(2). Necessary condition of coincidence of E-2 with the edge of regression on Sigma(2) is provided. (C) 1998 Elsevier Science S.A.