In this article we present a drastic dimension reduction method to link the biophysical parameters of an electromechanical model of the heart with a compact representation of cardiac motion. Our approach relies on a projection of the displacement fields along the whole cardiac motion to the space of reduced-polyaffine transformations. Using these transformations, not only we describe the motion using a very small number of parameters but we show that each of these parameters has a physiological meaning. Moreover, using a PLS regression on a learning set made of a large number of simulations, we are able to find which of the input parameters of the model most impact the motion and what are the main relations mapping the polyaffine representation to the parameters of the model. We illustrate the potential of this method for building a direct and very fast model characterized by a highly reduced number of parameters.