Area and volume in non-Euclidean geometry

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Abstract

We give an overview old and recent results on areas and volumes in hyperbolic
and spherical geometries. First, we observe the known results about Heron’s and Ptolemy’s theorems. Then we present non-Euclidean analogues of the Brahmagupta’s theorem for a cyclic quadrilateral. We produce also hyperbolic and spherical versions of the Bretschneider’s formula for the area of a quadrilateral. We give hyperbolic and spherical analogues of the Casey’s theorem which is a generalization of the Ptolemy’s equation. We give a short historical review of volume calculations for non-Euclidean polyhedra. Then we concentrate
on recent results concerning Seidel’s problem on the volume of an ideal tetrahedron, Sforza’s formula for a compact tetrahedron in H3 or S3 and volumes of non-Euclidean octahedra with symmetries
Original languageEnglish
Title of host publicationEIGHTEEN ESSAYS IN NON-EUCLIDEAN GEOMETRY
Editors Alberge, A Papadopoulos
PublisherEUROPEAN MATHEMATICAL SOC
Pages151-189
Number of pages39
ISBN (Print)978-3-03719-196-5
DOIs
Publication statusPublished - 2019

Publication series

NameIRMA Lectures in Mathematics and Theoretical Physics
PublisherEUROPEAN MATHEMATICAL SOC
Volume29

Keywords

  • CYCLIC POLYGONS
  • SEIDEL PROBLEM
  • FORMULA
  • THEOREM

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