Hyperbolic trigonometry is a basic tool in various studies, and various approaches are known(cf. e.g. Beardon , Fenchel , Meschkowski , Perron , Rees , Thurston ). This chapter gives an account based on the hyperboloid model which has grown out of discussions with Patrick Eberlein and Klaus-Dieter Semmler. In the first part we use the isometry group acting on the hyperboloid model to obtaing the trigonometric formulae of the triangle by comparing matrix elements. In the second part, beginning with Section 4, we generalize this to hexagons and similar configurations. Then we briefly sidestep to variable curvature. In the final part, Section 6, we describe a variant of the approach which is from Semmler  and uses a vector product and quaternions.
KeywordsVector Product Hyperbolic Plane Quaternion Algebra Geodesic Triangle Generalize Triangle
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