Given a set of points in the plane, we show that the θ-graph with 5 cones is a geometric spanner with spanning ratio at most √ 50 + 22 √ 5 ≈ 9.960. This is the first constant upper bound on the spanning ratio of this graph. The upper bound uses a constructive argument that gives a (possibly self-intersecting) path between any two vertices, of length at most √ 50 + 22 √ 5 times the Euclidean dis...

This paper discusses issues concerning the validation of solutions of construction problems in Dynamic Geometry Environments (DGEs) as compared to classic paper-and-pencil Euclidean geometry settings. We begin by comparing the validation criteria usually associated with solutions of construction problems in the two geometry worlds – the ‘drag test’ in DGEs and the use of only straightedge and c...

Non-Euclidean method of the generalized geometry construction is considered. According to this approach any generalized geometry is obtained as a result of deformation of the proper Euclidean geometry. The method may be applied for construction of space-time geometries. Uniform isotropic space-time geometry other, than that of Minkowski, is considered as an example. The problem of the geometric...

In two papers titled On the so-called non-Euclidean geometry, I and II ([32] and [34]), Felix Klein proposed a construction of the spaces of constant curvature -1, 0 and and 1 (that is, hyperbolic, Euclidean and spherical geometry) within the realm of projective geometry. Klein’s work was inspired by ideas of Cayley who derived the distance between two points and the angle between two planes in...

For more than two millennia, ever since Euclid’s geometry, the so called Archimedean Axiom has been accepted without sufficiently explicit awareness of that fact. The effect has been a severe restriction of our views of space-time, a restriction which above all affects Physics. Here it is argued that, ever since the invention of Calculus by Newton, we may actually have empirical evidence that t...

This will be a description of a few highlights in the early history of non-euclidean geometry, and a few miscellaneous recent developments. An Appendix describes some explicit formulas concerning volume in hyperbolic 3-space. The mathematical literature on non-euclidean geometry begins in 1829 with publications by N. Lobachevsky in an obscure Russian journal. The infant subject grew very rapidl...

Introduction 3 1. Affine geometry 4 1.1. Affine spaces 5 1.1.1. Euclidean geometry and its isometries 5 1.1.2. Affine spaces 7 1.1.3. Affine transformations 8 1.1.4. Tangent spaces 9 1.1.5. Acceleration and geodesics 10 1.1.6. Connections 11 1.2. The hierarchy of structures 11 1.3. Affine vector fields 12 1.4. Affine subspaces 13 1.5. Volume in affine geometry 14 1.6. Centers of gravity 14 1.7....

A Clifford algebra has three major multiplications: inner product, outer product and geometric product. Accordingly, the same Clifford algebra has three versions: Clifford vector algebra, which features on inner products and outer products of multivectors; Clifford bracket algebra, which features on pseudoscalars and inner products of vectors; Clifford geometric algebra, which features on geome...

The goal of this paper is to introduce and study analogues of the Euclidean Funk and Hilbert metrics on open convex subsets Ω of hyperbolic or spherical spaces. At least at a formal level, there are striking similarities among the three cases: Euclidean, spherical and hyperbolic. We start by defining non-Euclidean analogues of the Euclidean Funk weak metric and we give three distinct representa...

Introduction. This paper is concerned with several disconnected developments in distance geometry. §1 deals with the congruent imbedding of metric spaces in euclidean or Hubert spaces. By showing that the validity of the Pythagorean theorem insures the essentially euclidean character of the metric, the basic role this theorem plays in euclidean geometry is seen to be fully justified. In §2 the ...