In this work we analyse the evolution of the critical points of an image by the curvature and affine morphological scale spaces. We define the notions of circular and elliptic extremum and show that an extremum becomes circular by the curvature scale space and elliptic by the affine morphological scale space. The evolution of a saddle point by the curvature scale space is also described. And we show how these properties can lead to numerical methods for the simulation of the curvature scale space.