 Angle: Conformal GeometryJost-Hinrich Eschenburg
Chapter 6 in Geometry - Intuition and Concepts, 2022, pp 95-105 from Springer Abstract: 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 . Date: 2022 There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it. Related works: Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-658-38640-5_6 Ordering information: This item can be ordered from DOI: 10.1007/978-3-658-38640-5_6 Access Statistics for this chapter More chapters in Springer Books from Springer |
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