The conformal approach to Euclidean geometry introduced by David Hestenes, uses null vectors in an enlarged space to represent points. Here we show how these same techniques can be extended to the curved spacetimes relevant in cosmology. An extended example is given of the transfer of a result from 2d non-Euclidean geometry to 4d de Sitter space (the Origin Lemma in the Poincaré disc), and impl...

The spacetime ontology is considered in General Relativity (GR) in view of the choice of a frame of reference (FR). Various approaches to a description of the FR, such as coordinate systems, monads and tetrads are reviewed. It is shown that any of the existing FR definitions require a preexisting background spacetime, which, if defined independently of the FR, renders the spacetime absolute in ...

This paper, we define the Mus-Gradient metric on tangent bundle $TM$ by a deformation non-conform of Sasaki over an n-dimensional Riemannian manifold $(M, g)$. First investigate geometry and characterize new class proper biharmonic maps. Examples maps are constructed when all factors Euclidean spaces.

a lecture that nearly did not occur, Georg Friedrich Bernhard Riemann (1826–1866) proposed a visionary concept for the study of space [223, pp. 132–133]. To obtain the position of an unsalaried lecturer (Privatdozent) in the German university system, Riemann was required to submit an inaugural paper (Habilitationsschrift) as well as to present an inaugural lecture (Habilitationsvortrag). The to...

Spacelike constant mean curvature surfaces in Minkowski 3-space L have an infinite dimensional generalized Weierstrass representation. This is analogous to that given by Dorfmeister, Pedit and Wu for constant mean curvature surfaces in Euclidean space, replacing the group SU(2) with SU(1, 1). The non-compactness of the latter group, however, means that the Iwasawa decomposition of the loop grou...

Voronoi diagrams are fundamental data structures in computational geometry with several applications on different fields inside and outside computer science. This paper shows a CUDA algorithm to compute Voronoi diagrams on a 2D image where the distance between points cannot be directly computed in the euclidean plane. The proposed method extends an existing Dijkstra-based GPU algorithm to treat...

There is one Information Retrieval model that uses geometrical space: it is the Vector Space Model, which is defined in Euclidean Space. The paper shows that it is possible to define a Vector Space Model in non-Euclidean Space, too. Namely, the paper proposes a Vector Space Model over the Cayley-Klein Hyperbolic Geometry using a similarity measure derived from the hyperbolic distance. It is sho...

Representing videos by densely extracted local space-time features has recently become a popular approach for analysing actions. In this paper, we tackle the problem of categorising human actions by devising Bag of Words (BoW) models based on covariance matrices of spatio-temporal features, with the features formed from histograms of optical flow. Since covariance matrices form a special type o...

We present constructions inspired by the Ma-Schlenker example of [9] that show the non-rigidity of spherical inversive distance circle packings. In contrast to the use in [9] of an infinitesimally flexible Euclidean polyhedron, embeddings in de Sitter space, and Pogorelov maps, our elementary constructions use only the inversive geometry of the 2-sphere.

This paper proposes a method to improve the shortest path problem on a NURBS (Non-uniform rational basis spline) surfaces. It comes from an application of the theory in classic differential geometry on surfaces and can improve the distance problem not only on surfaces but in the Euclidean 3-space R . Keywords—shortest paths, geodesic-like method, NURBS surfaces.