Analytical expression for the shape of the rounded pyramid. I'm searching for an analytical equation approximating the pyramid with rounded tip. 
In particular, I have a pyramid whose base is an equilateral triangle with side "a", and height "h". The tip of the pyramid is rounded by some radius "r". (The point is to model the shape of the Berkovich indenter).
Could you suggest any (possibly simple) function that could approximate the shape of this pyramid?

Since I was not satisfied with existing formulations, I decided to approximate the pyramid with the rounded cone:

where a describes the slope of the cone's side and delta is the rounding parameter. Inside the square root it is reponsible for roundness, outside for having a tip of the approximated indenter in the origin (0,0,0).
This is a very crude approximation, however I have shown that it led to satisfactory results in my case.
If you ever find this information useful and use it in your publication, please cite:
K. Frydrych, CRYSTAL PLASTICITY FINITE ELEMENT SIMULATIONS OF THE INDENTATION TEST, Computer Methods in Materials Science, 19(2), 41-49, 2019.
https://www.researchgate.net/publication/337768801_CRYSTAL_PLASTICITY_FINITE_ELEMENT_SIMULATIONS_OF_THE_INDENTATION_TEST
