Universal Hyperbolic Geometry I: Trigonometry

广义双曲空间和广义de Sitter空间中的三角

Trigonometry in extended hyperbolic space and extended de Sitter space

本文研究了以双曲空间为子集的广义双曲空间中的双曲余弦和正弦律，它是双曲空间的解析延拓。研究了以deSitter空间$S^n_1$为子集的推广的deSitter空间中的球面余弦和球面正弦律，推广的deSitter空间也是deSitter空间的解析延拓。实际上，推广的双曲空间和推广的de Sitter空间是同一个空间，只是度量相差-1倍。因此，这两个扩展空间清楚地表明，并显然解释了为什么许多相应的公式在双曲和球面空间是非常相似的。从这些推广的三角学定律出发，我们可以对位于$S^2_1$上的双曲多边形的长度和角度之间的各种关系以及de Sitter多边形的关系给出一个连贯的、几何上简单的解释。

部分截断三角形与四面体的三角学

Trigonometry of partially truncated triangles and tetrahedra

本文利用双曲面模型和洛伦兹几何，建立了具有理想顶点的四边形和五边形的双曲余弦与正弦定理，并证明了部分截断四面体的横向长度仅取决于内部边长集合，且给出了界限。

仿射和射影泛几何

Affine and projective universal geometry

通过在纯代数环境中重铸度量几何，欧几里得几何和非欧几里得几何都可以在具有任意二次型的一般域上进行研究。这一新理论的仿射和投影版本在这里介绍，和主要公式的理性三角在平面上的延伸。这给出了一个统一的，计算模型的球形和双曲几何，允许扩展的许多结果欧几里德几何的相对论设置，并提供了一个新的度量方法代数几何。

Polygons in hyperbolic geometry 1: Rigidity and inversion of the n-inequality

Certain topics on polygons are extended from Euclidean to hyperbolic geometry. This first part deals with uniqueness and existence of cocyclic polygons with prescribed sidelengths. The non-Euclidean versions are more difficult due to the existence of three different types of circles in the hyperbolic plane. The second part will be concerned with the problem of maximizing the area of polygons with fixed sidelengths. By the metric character of these problems it is necessary to dispense to a certain extent with classical hyperbolic trigonometry because angles must be replaced by length quantities…

On Lobachevsky's trigonometric formulae

We elaborate on some important ideas contained in Lobachevsky's Pangeometry and in some of his other memoirs. The ideas include the following: (1) The trigonometric formulae, which express the dependence between angles and edges of triangles, are not only tools, but they are used as the basic elements of any geometry. In fact, Lobachevsky developed a large set of analytical and geometrical theorems in non-Euclidean geometry using these formulae. (2) Differential and integral calculus are developed in hyperbolic space without the use of any Euclidean model of hyperbolic space. (3) There exist m…

Universal Hyperbolic Geometry II: A pictorial overview

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.