be successfully used in multi-asperity contact models [@greenwood1985elliptic] instead of treatment elliptic contacts [@bush1975w; @johnson1985b], without any significant loss in accuracy. However, as seen by “naked eye” from Fig. \[fig:asperities\], a considerable population of asperities has a pronounced ellipticity. Nevertheless, the mechanical analysis can be successfully carried out using simple geometric mean curvature and associated correction factors even for strongly elliptic asperities, see Table 2 in [@greenwood2006w].

It seems that a good test for the surface generator would be a verification of agreement between asperities’ statistics detected on the surface and Greenwood’s analysis [@greenwood2006w], who obtained the following equation for the joint probability density of the ensemble of asperities’ principal curvatures[^4]: $$P(\kappa'_1,\kappa'_2) = \frac{27}{16\sqrt\pi}\kappa_1'\kappa_2'|\kappa_1'-\kappa_2'|\exp\left[-\frac{3}{16}\left(3\kappa_1^{\prime 2} + 3\kappa_2^{\prime 2} - 2\kappa'_1\kappa'_2\right)\right],
  \label{eq:Greenwood}$$ where $\kappa_i' = \kappa_i/\sqrt{m_4}$ represent normalized curvatures. This comparison is shown in Fig. \[fig:asp\_stat\]. Numerically, the statistics is performed over an ensemble of asperities detected on a single realization of a surface with a rich spectrum $L/\lambda_l = 4$, $L/\lambda_s=2048$, $L/\Delta x = 8192$ and $H=0.8$; in total this surface contains $\approx 580\,000$ asperities. In accordance with Greenwood’s findings, the most probable asperities possess the following ratio of principal curvatures $\kappa_1/\kappa_2 \approx 2.214$; the probability to find an asperity 