Universal hyperbolic geometry I: trigonometry

Wildberger, N. J.

doi:10.1007/s10711-012-9746-9

Cited by 21 publications

(27 citation statements)

References 19 publications

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“…This is a fundamental understanding. [135] So, in addition to Luneburg results, we claim that the detailed human perception experience is Hyperbolic Geometry based. More precisely, external world real system physical manifestation properties and related human perception are Hyperbolic Geometry representation based, while Euclidean approximated locally in the near-field.…”

Section: Anthropocentric Worldviewsupporting

confidence: 59%

How Random is Your Tomographic Noise? A Number Theoretic Transform (NTT) Approach

Fiorini

2014

Fundamenta Informaticae

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Discrete Tomography (DT), differently from GT and CT, focuses on the case where only few specimen projections are known and the images contain a small number of different colours (e.g. black-and-white). A concise review on main contemporary physical and mathematical CT system problems is offered. Stochastic vs. Combinatorially Optimized Noise generation is compared and presented by two visual examples to emphasise a major double-bind problem at the core of contemporary most advanced instrumentation systems. Automatic tailoring denoising procedures to real dynamic system characteristics and performance can get closer to ideal self-registering and selflinearizing system to generate virtual uniform and robust probing field during its whole designed service life-cycle. The first attempt to develop basic principles for system background low-level noise source automatic characterization, profiling and identification by CICT, from discrete system parameter, is presented. As a matter of fact, CICT can supply us with cyclic numeric sequences perfectly tuned to their low-level multiplicative source generators, related to experimental high-level overall perturbation (according to high-level classic perturbation computational model under either additive or multiplicative perturbation hypothesis). Numeric examples are presented. Furthermore, a practical NTT example is given. Specifically, advanced CT system, HRO and Mission Critical Project (MCP) for very low Technological Risk (TR) and Crisis Management (CM) system will be highly benefitted mostly by CICT infocentric worldview. The presented framework, concepts and techniques can be used to boost the development of next generation algorithms and advanced applications quite conveniently.

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Section: Anthropocentric Worldviewsupporting

confidence: 59%

How Random is Your Tomographic Noise? A Number Theoretic Transform (NTT) Approach

Fiorini

2014

Fundamenta Informaticae

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“…This conic is called the absolute in Cayley-Klein geometry. In this universal hyperbolic geometry (UHG), developed in [9][10][11][12], we take it to be a circle, typically in blue, and call it the null circle.…”

Section: The Polarity Of a Conic Discovered By Apolloniusmentioning

confidence: 99%

“…Due to the modern familiarity with linear algebra, it may be useful to reframe the projective setup above using homogeneous coordinates, where we follow: [9]. In a three-dimensional vector space of row vectors (x, y, z), we may define a (hyperbolic) point a ≡ [x : y : z] to be a one-dimensional subspace through a non-zero vector (x, y, z) .…”

Section: The Algebraic Approachmentioning

confidence: 99%

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Universal Hyperbolic Geometry, Sydpoints and Finite Fields: A Projective and Algebraic Alternative

Wildberger

2018

Universe

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Universal hyperbolic geometry gives a purely algebraic approach to the subject that connects naturally with Einstein's special theory of relativity. In this paper, we give an overview of some aspects of this theory relating to triangle geometry and in particular the remarkable new analogues of midpoints called sydpoints. We also discuss how the generality allows us to consider hyperbolic geometry over general fields, in particular over finite fields.Keywords: rational trigonometry; universal hyperbolic geometry; sydpoints; finite fields Two Famous Questions and a Projective/Algebraic Look at Hyperbolic GeometryWhile physicists have long pondered the question of the physical nature of the "continuum", mathematicians have struggled to similarly understand the corresponding mathematical structure. In the last decade, we have seen the emergence of rational trigonometry [1,2] as a viable alternative to traditional geometry, built not over a continuum of "real numbers", but rather algebraically over a general field, so also over the rational numbers, or over finite fields.Universal hyperbolic geometry (UHG) extends this understanding to the projective setting, yielding a new and broader approach to the Cayley-Klein framework (see [3]) for the remarkable geometry discovered now almost two centuries ago by Bolyai, Gauss and Lobachevsky as in [4][5][6]. See also [7,8] for the classical and modern use of projective metrical structures in geometry. In this paper, we will give an outline of this new approach, which connects naturally to the relativistic geometry of Lorentz, Einstein and Minkowski and also allows us to consider hyperbolic geometries over general fields, including finite fields.To avoid technicalities and make the subject accessible to a wider audience, including physicists, we aim to describe things both geometrically in a projective visual fashion, as well as algebraically in a linear algebraic setting. The Polarity of a Conic Discovered by ApolloniusWe augment the projective plane, which we may regard as a two-dimensional affine plane and a line at infinity, with a fixed conic. This conic is called the absolute in Cayley-Klein geometry. In this universal hyperbolic geometry (UHG), developed in [9-12], we take it to be a circle, typically in blue, and call it the null circle.The polarity associated with a conic was investigated by Apollonius and gives a duality between points a and lines A = a ⊥ in the space, which we also write as a = A ⊥ . Given a point a, consider any two lines through a, which meet the conic at two points each as in Figure 1. The other two diagonal points of this cyclic quadrilateral defines the dual line A. Remarkably, this construction does not depend on the choice of lines through a, as Apollonius realized.

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“…• Linear approach to elementary geometric objects such as geodesics, totally geodesic spaces, and bisectors [ChG], [Gir], [Hsi1], [Hsi2], [Wil1], [Wil2];…”

Section: Classic Geometries: Introduction Definition Examples and mentioning

confidence: 99%

Coordinate-Free Classic Geometries

Anan’in¹,

Grossi

2011

MMJ

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This paper is devoted to a coordinate-free approach to several classic geometries such as hyperbolic (real, complex, quaternionic), elliptic (spherical, Fubini-Study), and lorentzian (de Sitter, anti de Sitter) ones. These geometries carry a certain simple structure that is in some sense stronger than the riemannian structure. Their basic geometrical objects have linear nature and provide natural compactifications of classic spaces. The usual riemannian concepts are easily derivable from the strong structure and thus gain their coordinate-free form. Many examples illustrate fruitful features of the approach. The framework introduced here has already been shown to be adequate for solving problems concerning particular classic spaces.2000 Mathematics Subject Classification. 53A20 (53A35, 51M10).

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Universal hyperbolic geometry I: trigonometry

Cited by 21 publications

References 19 publications

How Random is Your Tomographic Noise? A Number Theoretic Transform (NTT) Approach

How Random is Your Tomographic Noise? A Number Theoretic Transform (NTT) Approach

Universal Hyperbolic Geometry, Sydpoints and Finite Fields: A Projective and Algebraic Alternative

Coordinate-Free Classic Geometries

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