  This paper is devoted to the classification problems concerning extended
deformations of convex polyhedra and real hyperplane arrangements in the
following senses: combinatorial equivalence of face posets, normal equivalence
on normal fans of convex polyhedra, and sign equivalence on half-spaces. The
extended deformations of convex polyhedra arise from parallel translations of
given half-spaces and hyperplanes, whose normal vectors give rise to the
so-called ``derived arrangement'' proposed by Rota as well as Crapo in
different forms. We show that two extended deformations of convex polyhedra are
normally (combinatorially, as a consequence) equivalent if they are
parameterized by the same open face of the derived arrangement. Note that these
extended deformations are based on parallel translations of the given
hyperplanes. It allows us to study three deformations of real hyperplane
arrangements: parallel translations, conings, and elementary lifts, whose
configuration spaces are parameterized by open faces of the derived
arrangement. Consequently, it gives a characterization of the normal,
combinatorial, and sign equivalences of those three deformations via the
derived arrangement. Additionally, the relationships among these three
equivalence relations are discussed, and several new descriptions of real
derived arrangements associated with faces and sign vectors of real hyperplane
arrangements are provided.
