Hyperbolic Geometry

Jürgen Richter-Gebert ()
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Jürgen Richter-Gebert: TU München, Zentrum Mathematik (M10) LS Geometrie

Chapter 25 in Perspectives on Projective Geometry, 2011, pp 483-503 from Springer

Abstract: Abstract Back to mathematics! This chapter is dedicated to several interesting topics in hyperbolic geometry. With our previous knowledge on the real projective plane RP2, on the complex projective line CP1, and of Cayley-Klein geometries we have an ideal departure point to explain several hyperbolic effects from an elegant and advanced standpoint. In particular, we will work out the relations between the Beltrami-Klein model and the Poincaré disk model. Compared to the general considerations in Chapters 20–23 we are now in a somewhat better situation. When we dealt with general Cayley-Klein geometries we spent a lot of our efforts on the treatment of case distinctions that arose from the various degrees of degeneracy of the fundamental conic. The algebraic structure became easier the less degenerate the fundamental conic was. Now we will deal only with one particular Cayley-Klein geometry, which in addition is nondegenerate.

Date: 2011
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DOI: 10.1007/978-3-642-17286-1_25

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