Mathematics – Geometric Topology
Scientific paper
2011-01-25
Mathematics
Geometric Topology
51 pages, 2 figures. Some additions and revisions, most extensively in Section 6
Scientific paper
A polygon in the hyperbolic plane is cyclic if a single circle contains all of its vertices; we will say it is "centered" if in addition its interior contains the center of this circle. We give necessary and sufficient conditions for a set of real numbers to be the side length collection of a cyclic or centered polygon. A cyclic polygon is uniquely determined by its collection of side lengths; its vertex angles vary as C^1 functions of side lengths; and so does the radius of the circle containing its vertices. We describe the derivatives of these quantities and show in consequence that the area of centered polygons is "monotonic" in side lengths. This does not hold for non-centered cyclic polygons, but for these the "horocyclic" polygons, those with all vertices on a horocycle, can be used to bound area below.
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