Parabolic, ridge and sub-parabolic curves on implicit surfaces with singularities

Abstract

We study parabolic, ridge and sub-parabolic curves on implicit surfaces defined by smooth functions $\mathcal{R}$-equivalent to $A_1^-$-singularity. To investigate ridge and sub-parabolic curves, we present the local parameterizations of the implicit surfaces, and we show the asymptotic behavior of the principal curvatures and directions by using the parameterization. We also present height and distance squared functions on implicit surfaces in the appendix.