The aim of this paper is to develop some techniques for automation of the mappings (between working and reference domains) required by reduced basis methods: the development of geometry mappings is indeed often a substantial impediment to the implementation of reduced basis techniques, especially in the context of the reduced basis element method (RBEM) and the reduced basis component method (RBCM). In the RBCM context, the geometry mappings are applied at the level of components. The methods have been tested on various cases to understand the limits of the approach and try to foresee and overcome the possible failures.