Some Classes of Frontals and Its Representation Formulas

In this paper, we characterize the extendibility of the normal curvature at singularities of frontals and give a representation formula for the class of frontals with this property. We introduce the relative normal curvature, which allows us to study the classical normal curvature, the asymptotic curves and the lines of curvature through singularities. Also, we provide representation formulas for wavefronts near all types of singularities and subclasses, such as wavefronts with extendable Gaussian curvature, bounded Gaussian curvature, and extendable principal curvature, among others.