In this two-part, paper, we present a model for the solidification of semicrystalline thermoplastic polymers that links the microscopic and macroscopic length scales. The model accounts for the important physical phenomena occurring during spherulitic crystallization and predicts microstructural evolution. In this first part, we concentrate on the growth of individual spherulites. We present a fast and accurate model that predicts the shape of a spherulite as well as the path of its constitutive lamellae in any thermal situation. In part II of this paper we shall couple this new model for spherulite growth with a nucleation law and a macroscopic heat flow computation, thus enabling us to model realistic crystallization conditions.