  In this manuscript we provide a fully geometric formulation for the induced
electromagnetic fields and their corresponding constitutive relations in moving
media. To this end, we present the reader with a brief geometric summary to
show how vector calculus electromagnetic theory is embedded in the more general
language of differential forms. Then, we consider the class of \emph{metric}
constitutive relations describing the medium in which electromagnetic fields
propagate. We explicitly obtain the components of the induced fields in a
moving medium, as seen in the the lab \emph{rest} frame. This allows us to read
the expressions for the permitivity, permeability and magnetoelectric matrices
for the moving medium which, in turn, can be interpreted as a different
physical material from the lab point of view
