Elliptic coordinate system

About: Elliptic coordinate system is a research topic. Over the lifetime, 670 publications have been published within this topic receiving 11135 citations. The topic is also known as: elliptical coordinate system & elliptic coordinates.

Calculation of radial coordinates of polar-coordinate raster scan

Abstract: The calculation of r2 =x2 +y2, where x and y are the Cartesian coordinates of a raster-scanned space, is carried forward by accumulation of 2x+1 and 2y+1 terms. A read-only memory can then be used for table look-up of r, the radial coordinate of raster scan in polar coordinates.

Superintegrable classical Zernike system

Abstract: We consider the differential equation that Zernike proposed to classify aberrations of wavefronts in a circular pupil, as if it were a classical Hamiltonian with a non-standard potential. The trajectories turn out to be closed ellipses. We show that this is due to the existence of higher-order invariants that close into a cubic Higgs algebra. The Zernike classical system thus belongs to the class of superintegrable systems. Its Hamilton-Jacobi action separates into three vertical projections of polar coordinates of sphere, polar, and equidistant coordinates on half-hyperboloids, and also in elliptic coordinates on the sphere.

Handling of Singularity in Finite-Difference Time-Domain Procedure for Solving Maxwell's Equations in Cylindrical Coordinate System

Abstract: A consistent approach to determine the components of electric and magnetic fields at r = 0 for a cylindrical coordinate system in a finite-difference time-domain calculation procedure is proposed. It is based on the superposition of cylindrical and Cartesian coordinate systems and transformation between them. Simple formulas have been derived for calculating the electromagnetic field components at a coordinate center; they can be used for solving two-dimensional and three-dimensional problems in a cylindrical coordinate system. The proposed method is used for calculating the outflow of an electromagnetic wave from a circular waveguide into free space.

Lie theory and the wave equation in space–time. 5. R‐separable solutions of the wave equation ψtt−Δ3ψ=0

Abstract: A detailed classification is made of orthogonal coordinate systems for which the wave equation ψtt−Δ3ψ=0 admits an R‐separable solution. Only those coordinate systems are given which are not conformally equivalent to coordinate systems that have been found in previous articles. We find 106 new coordinates to give a total of 367 conformally inequivalent orthogonal coordinates for which the wave equation admits an R‐separation of variables.

Method and system for device-independent determination of coordinates of a point displayed by means of a microscope

Abstract: The invention relates to a method and a system for device-independent determination of coordinates of a point (P), displayed by means of a microscope, whereby, firstly, the device coordinates (x1, y1, z1), for the displayed reference point (E1), in a device-dependent coordinate system corresponding to the given object-related reference coordinates (X1, Y1, Z1) of at least one reference point in a DICOM-coordinate system and a transformation rule (Φ) for the conversion of device-dependent coordinates (x, y, z) into the coordinates (X, Y, Z) of the DICOM-coordinate system are determined. Finally, to complete the device-independent coordinate determination, the device coordinates (xp, yp, zp) for a displayed point (P) are converted into device-independent coordinates (Xp, Yp, Zp) of the DICOM-coordinate system, by means of the determined transformation rule (Φ).