close-to-spherical is marginal. The overall mean geometrical curvature for the ensemble of asperities can be obtained from  by integration $$\bar\kappa'_g = \int\limits_0^\infty\int\limits_0^\infty \sqrt{\kappa_1'\kappa_2'} P(\kappa'_1,\kappa'_2) \,d\kappa_1'd\kappa_2' \approx 1.356
  \label{eq:mean_kg}$$

![\[fig:asp\_stat\]Comparison of the joint probability density computed numerically (on the left) and analytically, following Greenwood’s equation ; principal curvatures $\kappa_1,\kappa_2$ are normalized by the average algebraic curvature $\bar\kappa_a$.](Greenwood_statistics){width="100.00000%"}

\[sec:2\]Numerical model
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The numerical model is based on a deterministic multi-asperity Hertzian contact with included long range elastic interaction between asperities. A set of asperities $\mathcal I$ with in-plane coordinates $x_i,y_i$, summit height $z_i$ and curvature radius $r_i$ is distributed on the top of an elastic half-space. The asperity parameters come directly from the generated rough surfaces. This set of elastic asperities comes in contact with a rigid sphere of radius $R$. The sphere’s tip is located at $\{X_0,Y_0,Z_0\}$. Then the penetration between the sphere and $i$-th asperity is determined by: $$g_i = z_i  - \Delta z_i +\sqrt{R^2-(X_0-x_i)^2-(Y_0 -y_i)^2} - R - Z_0 = g_{i0} - \Delta z_i ,
  \label{eq:pen}$$ where $\Delta z_i$ is the vertical displacement of the asperity induced by contact forces acting on other asperities of the surface, and $g_{i0} = z_i  +\sqrt{R^2-(X_0-x_i)^2-(Y_0 -y_i)^2} - R - Z_0$. If the 