Mapping Cartesian Coordinates into Emission Coordinates: some Toy Models

After briefly reviewing the relativistic approach to positioning systems based on the introduction of the emission coordinates, we show how explicit maps can be obtained between the Cartesian coordinates and the emission coordinates, for suitably chosen set of emitters, whose world-lines are supposed to be known by the users. We consider Minkowski space-time and the space-time where a small inhomogeineity is introduced (i.e. a small "gravitational" field), both in 1+1 and 1+3 dimensions.Comment: 13 pages, 7 figures, Accepted for publication in International Journal of Modern Physics

Extended Fermi coordinates

We extend the notion of Fermi coordinates to a generalized definition in which the highest orders are described by arbitrary functions. From this definition rises a formalism that naturally gives coordinate transformation formulae. Some examples are developped in which the extended Fermi coordinates simplify the metric components.Comment: 16 pages, 1 figur

Noncommuting spherical coordinates

Restricting the states of a charged particle to the lowest Landau level introduces a noncommutativity between Cartesian coordinate operators. This idea is extended to the motion of a charged particle on a sphere in the presence of a magnetic monopole. Restricting the dynamics to the lowest energy level results in noncommutativity for angular variables and to a definition of a noncommuting spherical product. The values of the commutators of various angular variables are not arbitrary but are restricted by the discrete magnitude of the magnetic monopole charge. An algebra, isomorphic to angular momentum, appears. This algebra is used to define a spherical star product. Solutions are obtained for dynamics in the presence of additional angular dependent potentials.Comment: 5 pages, RevTex4 fil

Multiinstantons in curvilinear coordinates

The 'tHooft's 5N-parametric multiinstanton solution is generalized to curvilinear coordinates. Expressions can be simplified by a gauge transformation that makes $\eta$-symbols constant in the vierbein formalism. This generates the compensating addition to the gauge potential of pseudoparticles. Typical examples (4-spherical, 2+2- and 3+1-cylindrical coordinates) are studied and explicit formulae presented for reference. Singularities of the compensating field are discussed. They are irrelevant for physics but affect gauge dependent quantities.Comment: LaTeX file, 17 page

An Introduction to Hyperbolic Barycentric Coordinates and their Applications

Barycentric coordinates are commonly used in Euclidean geometry. The adaptation of barycentric coordinates for use in hyperbolic geometry gives rise to hyperbolic barycentric coordinates, known as gyrobarycentric coordinates. The aim of this article is to present the road from Einstein's velocity addition law of relativistically admissible velocities to hyperbolic barycentric coordinates along with applications.Comment: 66 pages, 3 figure