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Euclidean Geometry

In: Perspectives on Projective Geometry

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  • Jürgen Richter-Gebert

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

Abstract

In this chapter we will merge two different worlds: CP1 and RP2. Both can be considered as representing a real two-dimensional plane. They have different algebraic structures, and they both represent different compactifications of the Euclidean plane: For RP2 we added a line at infinity. For CP1 we added a point at infinity. Both spaces have different weaknesses and strengths. In the first two parts of the book we learned that RP2 is very well suited for dealing, for instance, with incidences of lines and points, with conics in their general form, and with cross-ratios. We did not have a proper way to talk about circles, angles, and distances in RP2. The previous two chapters introduced CP1. This space was very good for dealing with cocircularity and also for dealing with angles. However, lines were poorly supported by CP1. They had to be considered circles with infinite radius, and they were not even projectively invariant objects.

Suggested Citation

  • Jürgen Richter-Gebert, 2011. "Euclidean Geometry," Springer Books, in: Perspectives on Projective Geometry, chapter 18, pages 329-347, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-17286-1_18
    DOI: 10.1007/978-3-642-17286-1_18
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