The Hyperbolic Number Plane (original) (raw)
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n-Dimensional hyperbolic complex numbers
Advances in Applied Clifford Algebras, 1998
Direct product rings have received relatively little attention, perhaps because they are sometimes labeled "trivial" [8, p.6]. Nevertheless, the 2-dimensional direct product ring of the reals, when expressed in the "hyperbolic basis", is analogous in many ways to the system of complex numbers and also has a physical interpretation. This prompted an exploratory foray into the world of n-dimensional direct product rings of the reals to see how much can be extended from the 2-dimensional case (see, e.g. ). Section 1 provides algebraic notation, up to the point of defining polar coordinates. Section 2 uses analysis to explore differentiability and conformality.
On the Dual Hyperbolic Numbers and the Complex Hyperbolic Numbers
Journal of Computer Science & Computational Mathematics
In this study, we will introduce arithmetical operations on dual hyperbolic numbers w x y ju jv and hyperbolic complex numbers w x iy ju ijv which forms a commutative ring. Then, we will investigate dual hyperbolic number valued functions and hyperbolic complex number valued functions. One can see that these functions have similar properties.
Universal Hyperbolic Geometry II: A pictorial overview
Arxiv preprint arXiv:1012.0880, 2010
This article provides a simple pictorial introduction to universal hyperbolic geometry. We explain how to understand the subject using only elementary projective geometry, augmented by a distinguished circle. This provides a completely algebraic framework for hyperbolic geometry, valid over the rational numbers (and indeed any field not of characteristic two), and gives us many new and beautiful theorems. These results are accurately illustrated with colour diagrams, and the reader is invited to check them with ruler constructions and measurements.
Hyperbolic trigonometry in two-dimensional space-time geometry
Il Nuovo Cimento B, 2003
By analogy with complex numbers, a system of hyperbolic numbers can be introduced in the same way: z=x+h*y with h*h=1 and x,y real numbers. As complex numbers are linked to the Euclidean geometry, so this system of numbers is linked to the pseudo-Euclidean plane geometry (space-time geometry). In this paper we will show how this system of numbers allows, by means of a Cartesian representation, an operative definition of hyperbolic functions using the invariance respect to special relativity Lorentz group. From this definition, by using elementary mathematics and an Euclidean approach, it is straightforward to formalize the pseudo-Euclidean trigonometry in the Cartesian plane with the same coherence as the Euclidean trigonometry.
Complex Kleinian Groups, 2012
Complex Hyperbolic Geometry Chapter 2. Complex Hyperbolic Geometry V − ∪ V 0 ∪ V + , where each of these sets consists of the points (Z, z n+1) ∈ C n+1 satisfying that Z 2 is respectively smaller, equal to or larger than |z n+1 |. We will see in the following section that the projectivisation of V − is an open (2n)-ball B in P n C , bounded by [V 0 ], which is a sphere. This ball B serves as model for complex hyperbolic geometry. Its full group of holomorphic isometries is PU(n, 1), the subgroup of PSL(n+1, C) of projective automorphisms that preserve B. This gives a second natural source of discrete subgroups of PSL(n + 1, C), those coming from complex hyperbolic geometry. We finish this section with some results about subgroups of PSL(n + 1, C) that will be used later in the text. Proposition 2.1.2. Let Γ ⊂ PSL(n + 1, C) be a discrete group. Then Γ is finite if and only if every element in Γ has finite order. This proposition follows from the theorem below (see [182, Theorem 8.29]) and its corollary; see [160], [41] for details. Theorem 2.1.3 (Jordan). For any n ∈ N there is an integer S(n) with the following property: If G ⊂ GL(n, C) is a finite subgroup, then G admits an abelian normal subgroup N such that card(G) ≤ S(n)card(N). Corollary 2.1.4. Let G be a countable subgroup of GL(3, C), then there is an infinite commutative subgroup N of G.
On Interpretation of Hyperbolic Angle
2020
Minkowski spaces have long been investigated with respect to certain properties and substructues such as hyperbolic curves, hyperbolic angles and hyperbolic arc length. In 2009, based on these properties, Chung et al. [3] defined the basic concepts of special relativity, and thus; they interpreted the geometry of the Minkowski spaces. Then, in 2017, E. Nesovic [6] showed the geometric meaning of pseudo angles by interpreting the angle among the unit timelike, spacelike and null vectors on the Minkowski plane. In this study, we show that hyperbolic angle depends on time, t. Moreover, using this fact, we investigate the angles between the unit timelike and spacelike vectors.
2013
Relativistic hyperbolic geometry is a model of the hyperbolic geometry of Lobachevsky and Bolyai in which Einstein addition of relativistically admissible velocities plays the role of vector addition. The adaptation of barycentric coordinates for use in relativistic hyperbolic geometry results in the relativistic barycentric coordinates. The latter are covariant with respect to the Lorentz transformation group just as the former are covariant with respect to the Galilei transformation group. Furthermore, the latter give rise to hyperbolically convex sets just as the former give rise to convex sets in Euclidean geometry. Convexity considerations are important in non-relativistic quantum mechanics where mixed states are positive barycentric combinations of pure states and where barycentric coordinates are interpreted as probabilities. In order to set the stage for its application in the geometry of relativistic quantum states, the notion of the relativistic barycentric coordinates that relativistic hyperbolic geometry admits is studied.