Related papers: The Second Postulate of Euclid and the Hyperbolic …
H. L. Skala (1992) gave the first elegant first-order axiom system for hyperbolic geometry by replacing Menger's axiom involving projectivities with the theorems of Pappus and Desargues for the hyperbolic plane. In so doing, Skala showed…
Conditions, related to the so-called bending problem are considered for hypersurfaces of a pseudo-Euclidean space. Corresponding theorems are proved.
The aim of this article is to give a survey of combination theorems occurring in hyperbolic geometry, geometric group theory and complex dynamics, with a particular focus on Thurston's contribution and influence in the field.
We study the symplectic geometry of the moduli space of closed n-gons with fixed side-lengths in hyperbolic 3-space. We prove that these moduli spaces have a symplectic structure coming from Poisson Lie theory. We construct completely…
A class of second-order differential equations commonly arising in physics applications are considered, and their explicit hypergeometric solutions are provided. Further, the relationship with the Generalized and Universal Associated…
We provide hyperbolic analogues of some classical theorems in spherical geometry due to Menelaus, Euler, Lexell, Ceva and Lambert. Some of the spherical results are also made more precise.
The representations of Clifford algebras and their involutions and anti-involutions are fully investigated since decades. However, these representations do sometimes not comply with usual conventions within physics. A few simple examples…
We derive the Laws of Cosines and Sines in the super hyperbolic plane using Minkowski supergeometry and find the identical formulae to the classical case, but remarkably involving different expressions for cosines and sines of angles which…
We introduce the notions of geometric height and graded (geometric) relative hyperbolicity in this paper. We use these to characterize quasiconvexity in hyperbolic groups, relative quasiconvexity in relatively hyperbolic groups, and convex…
The geometry of closed surfaces equipped with a Euclidean metric with finitely many conical points of arbitrary angle is studied. The main result is that the image of a non-closed geodesic has 0 distance from the set of conical points.…
Graph representation learning in Euclidean space, despite its widespread adoption and proven utility in many domains, often struggles to effectively capture the inherent hierarchical and complex relational structures prevalent in real-world…
A deep relationship [arXiv:2503.17816v1] between real linear second order ordinary differential equations $u''\left(x\right)+h\left(x\right)u\left(x\right)=0$, with differentiable $h(x)$, and two dimensional hyperbolic geometry is…
This paper investigates the relationship between a system of differential equations and the underlying geometry associated with it. The geometry of a surface determines shortest paths, or geodesics connecting nearby points, which are…
We describe the first-order variations of the angles of Euclidean, spherical or hyperbolic polygons under infinitesimal deformations such that the lengths of the edges do not change. Using this description, we introduce a vector-valued…
We present two new constructions in the usual euclidean plane. We only deal with 'Grecian Geometry', with this phrase we mean elementary geometry in the two-dimensional space R 2 . We describe and prove two propositions about 'projections'.…
In Euclidean geometry, the Pythagorean theorem is presented as an equation involving three squares. This paper explores how analogous expressions may be identified in spherical and hyperbolic geometries.
Recent research in representation learning has shown that hierarchical data lends itself to low-dimensional and highly informative representations in hyperbolic space. However, even if hyperbolic embeddings have gathered attention in image…
We define and study an extended hyperbolic space which contains the hyperbolic space and de Sitter space as subspaces and which is obtained as an analytic continuation of the hyperbolic space. The construction of the extended space gives…
This article describes an entirely algebraic construction for developing conformal geometries, which provide models for, among others, the Euclidean, spherical and hyperbolic geometries. On one hand, their relationship is usually shown…
While geometry with transcendental curves, like the Quadratrix of Hippias and the Spiral of Archimedes, played a significant role in our modern developments of geometry and algebra. The investigation has fallen off in the modern era despite…