  In view of solving problems of geometric realizability of polyhedra with
given geometric constraints, we describe the space of geometric realizations of
a simply-connected triangulated euclidean polyhedron in $\mathbb{R}^3$ up to
similarity in terms of the angles of its faces and the angles between its
faces. To do so we describe it as the set of its triangular faces glued
together correspondingly and as the set of the polyhedral cones that it defines
around its vertices. We recompute its dimension at smooth points modulo a
combinatorial lemma.
