Following a brief review of the history of the link between Einstein’s velocity addition law of special relativity and the hyperbolic geometry of Bolyai and Lobachevski, we employ the binary operation of Einstein’s velocity addition to introduce into hyperbolic geometry the concepts of vectors, angles and trigonometry. In full analogy with Euclidean geometry, we show in this article that the in...

Having based on the analysis of Pietronero and collaborators the galaxy number counts can be considered as an evidence of that the geometry of the universe is euclidean and that the K-corrections are absent. In such a universe measuring the cosmological redshift of the object and measuring the flux from the object are inconsistent. It is considered the model of the universe which describes the ...

For a set S of n lines labeled from 1 to n, we say that S supports an n-vertex planar graph G if for every labeling from 1 to n of its vertices, G has a straight-line crossing-free drawing with each vertex drawn as a point on its associated line. It is known from previous work [4] that no set of n parallel lines supports all n-vertex planar graphs. We show that intersecting lines, even if they ...

According to Felix Klein’s influential Erlanger program of 1872, geometry is the study of properties of a space, which are invariant under a group of transformations. In Klein’s framework, the familiar Euclidean geometry consist of n-dimensional Euclidean space and its group of isometries. In general, a geometry is a pair (G,X), where X is a (sufficiently nice) space and G is a (sufficiently ni...

In blended-neighbor interpolation of scattered data, a tensor field represents a model of spatial correlation that is both anisotropic and spatially varying. In effect, this tensor field defines a non-Euclidean metric, a measure of distance that varies with direction and location. The tensors may be derived from secondary data, such as images. Alternatively, when the primary data to be interpol...

We review a procedure of factorizing the Minkowski space Dirac operator over a~suitable superspace, discuss its Euclidean version and apply worked out formalism in case od an almost-commutative operator. The presented framework is attempt to reconcile non-commutative geometry supersymmetry.

Formulas about the side lengths, diagonal lengths or radius of the circumcircle of a cyclic polygon in Euclidean geometry, hyperbolic geometry or spherical geometry can be unified.

Theoretical and empirical work on the geometry of environmental knowledge is discussed. Certain patterns of distanc.e and directional estimates collected from humans have been interpreted as being due to non-metric or non-Euclidean spatial knowledge. I argue that attempts to determine this geometry are inconsistent with existing theoretical models of human knowledge storage and use. Spatial est...