The Poincare Model of Hyperbolic Geometry in an Arbitrary Real Inner Product Space and an Elementary Construction of Hyperbolic Triangles with Prescribed Angles

摘要

It is well known that in a hyperbolic triangle the sum of angles is less then π, see e.g. [5] where Walter Benz deals extensively with the two-dimensional case. Among others he states there that for given values α, β, γ > 0 with α+β+γ < π there is always a hyperbolic triangle with these angles. In his book [2] Benz describes euclidean and hyperbolic geometry in a unified manner, and furthermore, in an arbitrary, possibly infinite dimensional real inner product space (X, · ) of dimension at least 2. In this paper we show the usefulness of the “dimension free” concepts of [2] and we combine these with elementary geometric constructions to get hyperbolic triangles in the Poincar′e model with arbitrarily prescribed angles.