The Hyperbolic Number Plane

Sobczyk, Garret

doi:10.1080/07468342.1995.11973712

Cited by 138 publications

(87 citation statements)

References 2 publications

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“…As it can be noted in figs. (2)(3)(4), they have also other properties of the corresponding Euclidean circles, i.e., the inscribed hyperbola is tangent to the triangle sides (and not to the external straight line) and its centre is inside the triangle. As far as the ex-inscribed hyperbolas are concerned, their centres are outside the triangle and they are tangent to a triangle side and to two external straight lines.…”

Section: Discussionmentioning

confidence: 99%

“…(1). As it is well known, such units (both elliptic and hyperbolic) can be decomposed into idempotent basis ui: u+, u−, where u+ = 1/2(1 + u), u− = 1/2(1 − u), for which it results: (ui) n = ui and u+ u− = 0 [4,9]. Such decomposition is very useful for the effective calculation of the functions of hypercomplex variable [15] It has been shown [1,2], depending on the sign of the real quantity…”

Section: Basic Conceptsmentioning

confidence: 99%

“…In these representative planes, that are called with the names of the number systems, i.e., elliptic or hyperbolic, we associate the points P ≡ (x, y) with the general bidimensional hypercomplex numbers z = x + u y. Their topology is the same of the Euclidean plane for elliptic numbers, and of the pseudo-Euclidean (space-time) plane for hyperbolic numbers [13,4]. The hyperbolic plane is divided in four sectors [2] by the straight lines (null lines [3, pag. 179], a segment or line is said to be of the first (second) kind if it is parallel to a line through the origin located in the sectors containing the axis Ox (Oy) 5 .…”

Section: Geometrical Representationmentioning

confidence: 99%

“…Indeed their square module given by 1 |z| 2 = zz ≡ x 2 − y 2 is the Lorentz invariant of two-dimensional special relativity, and their unimodular multiplicative group is the special relativity Lorentz group [3,4]. These relations have been used to extend [5] and to generalise [6] the two-dimensional special relativity.…”

Section: Introductionmentioning

confidence: 99%

“…For parabolic and hyperbolic numbers the division is again not possible for numbers (z0) with zero module (see paragraph 2.2), but from this condition does not follow x = y = 0 and for these numbers there exist some non-zero numbers whose product with z0 is zero. Such numbers are called divisors of zero 4. This system has been widely studied by Yaglom[3], who has shown the link existing between the parabolic numbers and the Galileo group of classical kinematics.…”

mentioning

confidence: 99%

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Two-dimensional hypercomplex numbers and related trigonometries and geometries

Catoni

Cannata

Catoni

et al. 2004

AACA

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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; u 2 = 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.

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Section: Discussionmentioning

confidence: 99%