We present a consistent methodology for analyzing and computing both the tooth-flanks of hypoid gears and their geometrie contact in a gear drive. The to a great extend new approach is based on a very general model of the generating process of hypoid gears. The quantities of the main interest are constructively characterized as first-order singularities of suitable non linear functions. The closed analytical treatment of the matter allows in particular the efficient evaluation ofthe sensitivities of the characterized quantities with respect to arbitrary model parameters. These sensitivities constitute the final key to the application of modern numerical methods to optimize the tooth-flank geometry with respect to desired tooth-contact properties. To lay the rnathematical Inundation of all our investigations, we first examine certain generic singularities of functions that are known as fold points. Therefore, we apply an implicit l.iapunov Schmidt reduction as the central mathematical tool. In contrast to the many existing accounts on this technique, we here consider functions that are only once locally Lipschitz continuously differentiable. For that purpose we use certain elements of nonsrnooth analysis. The characterization and classification of singularities takes place in the domain of the functions, whereas the geometrie meaning transpires and will be discussed in the range space. The constructive approach leads to minimally augrnented defining systems that are suitable for the efficient and accurate computation of singular sets and subsequently oftheir images.