Barchans are crescent shaped sand dunes forming in aride regions with unidirectional wind and limited sand supply. We report analytical and numerical results for dune shapes under different environmental conditions as obtained from the so called minimal model of aeolian sand dunes. The profiles of longitudinal vertical slices i.e. along the wind direction are analyzed as a function of wind speed and sand supply. Shape transitions can be induced by changes of mass, wind speed and sand supply. Within a minimal extension of the model to the transverse direction the scale invariant profile of transverse vertical cuts can be derived analytically.