  We consider Riemannian 4d BF lattice gauge theory, on a triangulation of
spacetime. Introducing the simplicity constraints which turn BF theory into
simplicial gravity, some geometric quantities of Regge calculus, areas, and 3d
and 4d dihedral angles, are identified. The parallel transport conditions are
taken care of to ensure a consistent gluing of simplices. We show that these
gluing relations, together with the simplicity constraints, contain the
constraints of area-angle Regge calculus in a simple way, via the group
structure of the underlying BF gauge theory. This provides a precise road from
constrained BF theory to area-angle Regge calculus. Doing so, a framework
combining variables of lattice BF theory and Regge calculus is built. The
action takes a form {\it \`a la Regge} and includes the contribution of the
Immirzi parameter. In the absence of simplicity constraints, the standard spin
foam model for BF theory is recovered. Insertions of local observables are
investigated, leading to Casimir insertions for areas and 6j-symbols for 3d
angles. The present formulation is argued to be suitable for deriving spin foam
models from discrete path integrals.
