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Angle: Conformal Geometry

In: Geometry - Intuition and Concepts

Author

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  • Jost-Hinrich Eschenburg

    (Universität Augsburg, Institut für Mathematik)

Abstract

Distances can also be used to measure angles; for example, a triangle with side lengths 3, 4, 5 is right-angled (why?), which was used already by the ancient Egyptians in order to construct right angles. Conversely, however, distances cannot be determined only from angles. But there is a geometry with the angle as its only basic notion, called conformal geometry Geometry conformal ; it is much less known than metric geometry. Its “isomorphisms” are the conformal (i.e. angle-preserving) mappings. A great surprise is Liouville’s Theorem: In dimension 2 there is an infinite-dimensional family of conformal mappings, namely all holomorphic and antiholomorphic complex functions. But in dimension 3 (and higher), conformal mappings automatically preserve the set of spheres and planes in space; such mappings form a family with finitely many parameters and can easily be determined. To prove this we use the differential geometry developed in the previous chapter. Liouville’s Theorem allows us to study conformal geometry in space by considering the “space” of spheres and planes; this has its own metric structure which is related to the spacetime geometry of Special RelativityRelativity .

Suggested Citation

  • Jost-Hinrich Eschenburg, 2022. "Angle: Conformal Geometry," Springer Books, in: Geometry - Intuition and Concepts, chapter 6, pages 95-105, Springer.
    • Handle: RePEc:spr:sprchp:978-3-658-38640-5_6
    DOI: 10.1007/978-3-658-38640-5_6
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