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Advances in Applied Clifford Algebras

, Volume 14, Issue 1, pp 47–68 | Cite as

Two-dimensional hypercomplex numbers and related trigonometries and geometries

  • Francesco Catoni
  • Roberto Cannata
  • Vincenzo Catoni
  • Paolo Zampetti
Original Paper

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 \(\{ z = x + uy;\;u^2 = \alpha + u \beta;\;x, y, \alpha, \beta \in {\mathbf{R}};\;u \notin {\mathbf{R}}\} \) 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.

Keywords

Complex Number General System Number System Euclidean Geometry Plane Geometry 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Birkhäuser Verlag, Basel 2004

Authors and Affiliations

  • Francesco Catoni
    • 1
  • Roberto Cannata
    • 1
  • Vincenzo Catoni
    • 1
  • Paolo Zampetti
    • 1
  1. 1.ENEACentro Ricerche CasacciaRomaItaly

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