The level curves of an analytic function germ can have bumps (maxima of Gaussian curvature) at unexpected points near the singularity. This phenomenon is fully explored for f(z,w)∈C{z,w}, using the Newton–Puiseux infinitesimals and the notion of gradient canyon. Equally unexpected is the Dirac phenomenon: as c→0, the total Gaussian curvature of f=c accumulates in the minimal gradient canyons, and nowhere else. Our approach mimics the introduction of polar coordinates in Analytic Geometry.