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Incidence: Projective Geometry

In: Geometry - Intuition and Concepts

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  • Jost-Hinrich Eschenburg

    (Universität Augsburg, Institut für Mathematik)

Abstract

Projective geometry is the proper domain for the notion of incidence for straight lines and points; it knows no other basic notions. The basic ideas were developed 600 years ago with the discovery of central perspective. The artists pioneered the mathematicians. Just as in a perspective view parallel lines appear to have a point of intersection on the horizon, so parallelism is interpreted as “intersecting at infinity”. For this, geometry must be extended by “points at infinity”. These arise quite easily by embedding into linear algebra: This extended (“projective”) geometry, however, no longer takes place on a vector space as affine geometry does, but on the set of its one-dimensional linear subspaces. The structure-preserving transformations (“collineations”) are then simply the (semi-)linear isomorphisms of the vector space. Now for the first time in this book interesting geometric theorems are discussed, the theorems of Desargues, Brianchon and Pascal. We will get to know conic sections and quadrics, and at the end an important numerical quantity which is invariant under projective transformations: the cross-ratio.

Suggested Citation

  • Jost-Hinrich Eschenburg, 2022. "Incidence: Projective Geometry," Springer Books, in: Geometry - Intuition and Concepts, chapter 3, pages 21-55, Springer.
    • Handle: RePEc:spr:sprchp:978-3-658-38640-5_3
    DOI: 10.1007/978-3-658-38640-5_3
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