A big challenge in the last few years is to perform simulations of cardiovascular systems. Since it would be too computational expensive and complicated to simulate an entire blood circuit, an idea is to use a Domain Decomposition (DD) technique and to split it into many little parts. But these basic geometrical shapes can still differ quite a lot in forms and size. Therefore for each type of geometry we can define some very few reference geometries. Those reference geometries can then be deformed by a proper suitable map to any needed geometry. The aim of this report is to analyze the Transfinite Map (TM) and we introduce a possible extension of it. This deformations from the reference geometry to the computational one lead in general to a not affine map and affects strongly the complexity of the problem. In order to recover the affine decomposition we use the Empirical Interpolation Method that allows to approximate a not affine function with an affine one.