Area of polygons in hyperbolic geometry
Bailey, Benjamin Aaron
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Consider the Poincare model for hyperbolic geometry on the unit disc and an arbitrary n-gon in this geometry. Chapter I gives a brief introduction to the conformal metric, which generates hyperbolic geometry. In Chapter H, there is an analytic derivation of a convenient computational formula for the hyperbolic area of a hyperbolic n-gon in terms of the coordinates of its vertices, as well as an insightful geometric interpretation of this formula in terms of naturally occurring angles of the n-gon. Chapter III extends the formulas in Chapter II to the closed unit disc with an alternative, more geometrically motivated proof. Chapter IV uses the results of Chapter I to establish identities between hyperbolic area and perimeter of an n-gon. A proof of the existence of a solution for the following isoperimetric problem also is given: Maximize (if such a maximum exists) the area of an n-gon with fixed perimeter.