Indau, Johann

A discrete uniformization theorem for polyhedral surfaces

“…A theorem of Rivin [21] shows that D pl (T ) is a real analytic cell of dimension −3χ(S − V ). Indeed, one takes the open neighborhood of D pl (T ) to be P (T ) and fixes e 1 ∈ E. Define h to be the real analytic map sending x to (φ 0 (x), x(e 1 )) where φ 0 (x)(e) = α + α where α and α are angles facing e. Rivin proved that h is a real analytic diffeomorphism into an open subset of a codimension-1 affine subspace of R E × R so that h(D pl (T )) is a convex polytope and faces of D pl (T ) are subsets defined by α + α = π for some collection of edges e. By [2], [5], if W = D pl (T 1 ) ∩ ....D pl (T k ) = ∅, then W is a face of D pl (T i ) for each i. Indeed, W is the face of D pl (T i ) defined by the set of equalities: α + α = π for all edges e / ∈ ∩ k j=1 E(T j ).…”