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Two-dimensional hypercomplex numbers and related trigonometries and geometries

TitoloTwo-dimensional hypercomplex numbers and related trigonometries and geometries
Tipo di pubblicazioneArticolo su Rivista peer-reviewed
Anno di Pubblicazione2004
AutoriCatoni, F., Cannata R., Catoni V., and Zampetti P.
RivistaAdvances in Applied Clifford Algebras
Volume14
Paginazione47-68
ISSN01887009
Abstract

All the commutative hypercomplex number systems can be associated with a geometry. In two dimensions, by analogy with complex numbers, a general system of hypercomplex numbers z = x + u y; u2 = α + u β; x, y, α, β ∈ R; u ∉ R can be introduced and can be associated with plane Euclidean and pseudo-Euclidean (space-time) geometries. In this paper we show how these systems of hypercomplex numbers allow to generalise some well known theorems of the Euclidean geometry relative to the circle and to extend them to ellipses and to hyperbolas. We also demonstrate in an unusual algebraic way the Hero formula and Pytaghoras theorem, and show that these theorems hold for the generalised Euclidean and pseudo-Euclidean plane geometries. © 2004, Birkhauser Verlag AG. All rights reserved.

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cited By 18

URLhttps://www.scopus.com/inward/record.uri?eid=2-s2.0-33748923094&doi=10.1007%2fs00006-004-0008-2&partnerID=40&md5=636c512115eb149d509f175369990c26
DOI10.1007/s00006-004-0008-2
Citation KeyCatoni200447