\sqrt{\kappa_1\kappa_2} = 2\sqrt{ab - c^2}$$

![\[fig:asperities\]Generated rough surfaces (upper panel), detected elliptic asperities (middle panel) and approximate circular asperities with geometric mean curvature. For all surfaces, the Hurst exponent $H=0.8$, $L/\Delta x=2048$, lower cut-off $L/\lambda_l=4$, and root mean squared height $\sigma/L=0.025$: (a) $L/\lambda_s=16$, (b) $L/\lambda_s=32$, (c) $L/\lambda_s=64$. The surfaces are generated from the same white noise, therefore they are visually similar. The colour of asperities is selected according to the height of the summit.](asperities){width="100.00000%"}

In Fig. \[fig:asperities\] we show generated rough surfaces with different spectral content and the detected asperities approximated by ellipses with relevant orientation and circles with the geometric mean curvature. Of course, the ellipses and circles give only an approximate representation of the roughness: anyway, the Hertzian theory is valid for arbitrary elliptic paraboloids, which can be approximated by ellipsoids near their summits. The size of ellipses and circles corresponds to $L=1$ (l.u.[^3]) and root mean squared height (RMS height) $\sqrt{\langle (z-\bar z)^2\rangle}/L = \sigma/L =  0.025$. Obviously, a decrease of the RMS height would result in an increase of asperities’ radii. It was shown by Greenwood [@greenwood2006w], that according to Nayak’s theory [@nayak1971tasme], the asperities are only “mildly” elliptic, and thus, the geometric mean curvature can 