Computational geometry in Heisenberg group Heis^3
  Among eight possible geometric structures on three-dimensional manifolds less studied from the differential geometric point of view are those modelled on the Heisenberg group $Heis^3$. We consider the Heisenberg left-invariant metric and use some results on Levi-Civita connection and curvature tensor to present solutions of equations for geodesic lines in the Heisenberg group. Using "Mathematica" software package we also present drawings of geodesic lines and metric balls in the Heisenberg group. 
Andrey Marenich