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Cayley-Klein Geometries

In: Perspectives on Projective Geometry

Author

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  • Jürgen Richter-Gebert

    (TU München, Zentrum Mathematik (M10) LS Geometrie)

Abstract

We started out developing projective geometry for two reasons: It was algebraically nice and it helped us to get rid of the treatment of many special situations that are omnipresent in Euclidean geometry. Then, to express Euclidean geometry in a projective setup, we needed the help of complex numbers, our special points I and J, cross-ratios, and Laguerre’s formula. We now come to another pivot point in our explanations: We will see that our treatment of Euclidean geometry in a projective framework is only a special case of a variety of other reasonable geometries. One might ask what it means to be a geometry in that context. For us it means that there are notions of points, lines, incidence, distances, and angles with a certain reasonable interplay. Besides Euclidean geometry, among those geometries there are quite a few prominent examples, such as hyperbolic geometry, elliptic geometry, and relativistic space-time geometry.

Suggested Citation

  • Jürgen Richter-Gebert, 2011. "Cayley-Klein Geometries," Springer Books, in: Perspectives on Projective Geometry, chapter 20, pages 375-398, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-17286-1_20
    DOI: 10.1007/978-3-642-17286-1_20
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