Topic
Rational trigonometry
About: Rational trigonometry is a research topic. Over the lifetime, 46 publications have been published within this topic receiving 351 citations.
Papers
TL;DR: In this article, a new method to obtain trigonometry for real spaces of constant curvature and metric of any (even degenerate) signature is presented, which can be described as ''curvature/signature (in)dependent trigonometrics''.
Abstract: A new method to obtain trigonometry for the real spaces of constant curvature and metric of any (even degenerate) signature is presented. The method could be described as `curvature/signature (in)dependent trigonometry' and encapsulates trigonometry for all these spaces into a single basic trigonometric group equation. This brings to its logical end the idea of an `absolute trigonometry', and provides equations which hold true for the nine two-dimensional spaces of constant curvature and any signature. This family of spaces includes both relativistic and non-relativistic spacetimes; therefore a complete discussion of trigonometry in the six de Sitter, Minkowskian, Newton-Hooke and Galilean spacetimes follow as particular instances of the general approach. Distinctive traits of the method are `universality' and `self-duality': every equation is meaningful for the nine spaces at once, and displays invariance explicitly under a duality transformation relating the nine spaces amongst themselves. These basic structural properties allow a complete study of trigonometry and, in fact, any equation previously known for the three classical (Riemannian) spaces also has a version for the remaining six `spacetimes'; in most cases these equations are new.
91 citations
80 citations
Posted Content•
TL;DR: Hyperbolic geometry is developed in a purely algebraic fashion from first principles, without a prior development of differential geometry as discussed by the authors, and the natural connection with the geometry of Lorentz, Einstein and Minkowski comes from a projective point of view, with trigonometric laws that extend to ''points at infinity', here called ''null points', and beyond to ''ideal points' associated to a hyperboloid of one sheet''.
Abstract: Hyperbolic geometry is developed in a purely algebraic fashion from first principles, without a prior development of differential geometry. The natural connection with the geometry of Lorentz, Einstein and Minkowski comes from a projective point of view, with trigonometric laws that extend to `points at infinity', here called `null points', and beyond to `ideal points' associated to a hyperboloid of one sheet. The theory works over a general field not of characteristic two, and the main laws can be viewed as deformations of those from planar rational trigonometry. There are many new features.
35 citations
TL;DR: Hyperbolic geometry is developed in a purely algebraic fashion from first principles, without a prior development of differential geometry as mentioned in this paper, and the theory works over a general field not of characteristic two, and the main laws can be viewed as deformations of those from planar rational trigonometry.
Abstract: Hyperbolic geometry is developed in a purely algebraic fashion from first principles, without a prior development of differential geometry. The natural connection with the geometry of Lorentz, Einstein and Minkowski comes from a projective point of view, with trigonometric laws that extend to ‘points at infinity’, here called ‘null points’, and beyond to ‘ideal points’ associated to a hyperboloid of one sheet. The theory works over a general field not of characteristic two, and the main laws can be viewed as deformations of those from planar rational trigonometry. There are many new features; this paper gives 92 foundational theorems.
26 citations
29 Dec 2010
TL;DR: In this paper, a simple pictorial introduction to universal hyperbolic geometry is provided, using only elementary projective geometry, augmented by a distinguished circle, valid over the rational numbers (and indeed any field not of characteristic two).
Abstract: 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.
18 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2020 | 2 |
| 2019 | 6 |
| 2018 | 2 |
| 2017 | 1 |
| 2015 | 2 |
| 2014 | 1 |