is described in this way, it may not be clear what shape the resulting polyhedron has, what shapes its faces have, or even how many faces it has. Alexandrov's original proof does not lead to an algorithm for constructing the polyhedron (for instance by giving coordinates for its vertices) realizing the given metric space. In 2008, Bobenko and Izmestiev provided such an algorithm. Their algorithm can approximate the coordinates arbitrarily accurately, in pseudo-polynomial time. Related results One of the first existence and uniqueness theorems for convex polyhedra is Cauchy's theorem, which states that a convex polyhedron is uniquely determined by the shape and connectivity of its faces. Alexandrov's theorem strengthens this, showing that even if the faces are allowed to bend or fold, without stretching or shrinking, then their connectivity still determines the shape of the polyhedron. In turn, Alexandrov's proof of the existence part of his theorem uses a strengthening of Cauchy's theorem by Max Dehn to infinitesimal rigidity. An analogous result to Alexandrov's holds for smooth convex surfaces: a two-dimensional Riemannian manifold whose Gaussian curvature is everywhere positive and totals 4 can be represented uniquely as the surface of a smooth convex body in three dimensions. The uniqueness of this representation is a result of Stephan Cohn-Vossen from 1927, with some regularity conditions on the surface that were removed in later research. Its existence was proven by Alexandrov, using an argument involving limits of polyhedral metrics. Aleksei Pogorelov generalized both these results, characterizing the developments of arbitrary convex bodies in three dimensions. Another result of Pogorelov on the geodesic metric spaces derived from convex polyhedra is a version of the theorem of the three geodesics: every convex polyhedron has at least three simple closed quasigeodesics. These are curves that are locally straight lines except when they pass through a vertex, where they are required to have angles of less than on both sides of them. The developments of ideal hyperbolic polyhedra can be characterized in a similar way to Euclidean convex polyhedra: every two-dimensional manifold with uniform hyperbolic geometry and finite area, combinatorially equivalent to a finitely-punctured sphere, can be realized as the surface of an ideal polyhedron. References Geodesic (mathematics) Mathematics of rigidity Theorems in convex geometry Theorems in discrete geometry Uniqueness theorems