The joint histogram of second order scale space differential invariants of natural images (including textures) is typically clustered about parabolic surface patches, whereas symmetrical elliptical patches (local convexities or concavities) are very rare and symmetrical hyperbolical patches also occur less frequently than parabolic patches. We trace the origin of this striking effect in the context of Gaussian random noise. For this case one may derive the joint histogram of curvedness and shape index analytically. The empirical observations are fully corroborated. In deriving these results we introduce a polar coordinate system in the space of second order scale space derivatives that turns out to be particularly useful in the study of the statistics of local curvature properties. The empirical observations apply also to non-Gaussian noise (e.g., Brownian noise) as well as to photographs of natural scenes. We discuss general arguments that help explain these observations.