Lobachevsky geometry of (super)conformal mechanics

We give a simple geometric explanation for the similarity transformation mapping one-dimensional conformal mechanics to free-particle system. Namely, we show that this transformation corresponds to the inversion of the Klein model of Lobachevsky space (non-compact complex projective plane) CP˜1. We also extend this picture to the N=2kN=2k superconformal mechanics described in terms of Lobachevsky superspace CP˜1|k.