Planar polygons and hyperbolic polyhedra
The authors consider the space of polygons in the Euclidean plane having fixed prescribed angles. These form a vector space on which the measurement of oriented area defines a quadratic form. In the case of convex polygons this construction is used to associate a convex polyhedron in a suitable hyperbolic space with the given space. The relations between the structure of this polyhedron and the geometry of the corresponding polygons are investigated. This interpretation is used for another proof of the existence of Im Hof's hyperbolic Coxeter orthoschemes [see \textit{H.-C. Im Hof}, Expo. Math. 3, 179-186 (1985; Zbl 0572.51012), and Bull. Soc. Math. Belg. Ser. A 42, No. 3, 523-545 (1990)].
scientific article; zbMATH DE number 93783