In this article we explicitly construct Stäckel separable systems in separation coordinates with the help of separation curve as introduced by Sklyanin. Further, we construct explicit transformation bewteen separable and ‡at coordinates for ‡at Stäckel systems. We also exploit the geometric structre of these systems in the obtained ‡at coordinates. These coordinates generalize the well known ge...

This paper shows the coordinates influence on singularity of a three degree-of-freedom structure, namely, three-Universal-Prismatic-Spherical (3-UPS) parallel manipulator. Rotational coordinates, which are chosen to define the orientation of the platform, affect the singularity of the manipulator. Euler parameters, which don't have any inherent geometrical singularity are utilized, however they...

The generally adopted approach in theory of relativistic strings and membranes, is similar to use of Lagrange coordinates in continious media mechanics. One can use an alternative approach, which is similar to use of Euler coordinates. Under such approach the consideration of thick (delocalized) membranes is natural. Membrane kinematics, which coorespond to Euler coordinates is constructed. Var...

In this paper a noniterative method for the numerical generation of orthogonal curvilinear coordinates for plane annular regions between two arbitrary smooth closed curves has been developed. The basic generating equation is the Gaussian equation for a Euclidean space under the constraint of orthogonality. The resulting equation has been solved analytically for the case of isothermic coordinate...

In this section we discuss how to calibrate an overhead video camera used for tracking robots moving on a rectangular field. First, the radial distortion of the camera must be eliminated. The resulting corrected image represents a projective transformation of the field’s surface to the camera’s imaging plane. We show how to find the projective transformation, and how to recover from this inform...

Weyl-Heisenberg frames are complete signal representations corresponding to rectangular tiling of the time-frequency plane. Extensions of these frames are obtained in the rotated time-frequency planes by using the fractional Fourier transformation. It is shown that rotation does not affect the frame bounds. For some specific angles, lattices in rotated coordinates will map to the lattices in th...

Nonrelativistic Newton and Schrödinger equations remain correct not only under holonomic but also under nonholonomic transformations of the spacetime coordinates. Here we study the properties of transformations which are holonomic in the space coordinates while additionally tranforming the time in a path-dependent way. This makes them nonholonomic in spacetime. The resulting transformation form...

We present an efficient modal method to calculate the two-dimensional Green’s function for electromagnetics in curvilinear coordinates. For this purpose the coordinate transformation based differential method, introduced for the numerical analysis of surface-relief gratings, is directly used with perfectly matched layers (PMLs). The covariant formalism Maxwell’s equations, very convenient for t...

The 'tHooft's 5N-parametric multiinstanton solution is generalized to curvilinear coordinates. Expressions can be simplified by the gauge transformation that makes η-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 formul...