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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.


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TL;DR: In this article, a general analytic formula for the two-center Coulomb integrals over Slater-type orbitals in elliptical coordinates is obtained, expressed in terms of a product of the well-known auxiliary functions Ak(p) and Bk(p), and incomplete gamma functions.
Abstract: A general analytic formula is obtained for the two-center Coulomb integrals over Slater-type orbitals in elliptical coordinates. Finite series expansions are used in the evaluation of the radial part of the integrals. The analytic formula is expressed in terms of a product of the well-known auxiliary functions Ak(p) and Bk(p) and incomplete gamma functions. Recursive relations for the computer evaluation of these functions are given as well. The recursive relations are stable and our computer results are in good agreement with the benchmark values given in the literature. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2003

23 citations

Journal ArticleDOI
TL;DR: In this paper, a new type of field expansion for static or quasi-static (here magnetic) two-dimensional fields is proposed and investigated, which are particular solutions of the potential equation in plane elliptic coordinates obtained by the method of separation.
Abstract: Standard textbooks on beam dynamics study the impact of the magnetic field quality on the beam using field representations based on circular multipoles. Iron dominated magnets, however, typically provide a good field region with a non-circular aspect ratio (i.e. an ellipse whose axis a is significantly larger than the axis b); a boundary not ideal for circular multipoles. The development of superconductors, originally driven to reach fields above ≈ 2 T , allows using them today in completely different fields: iron dominated DC magnets, to save the energy for coil powering as well as repeatedly fast ramped magnets. The cold mass of magnets, housed in common cryostats sectors, makes it tedious to implement additional correction magnets at a later stage, as it requires to warm up the sections where the magnets should be installed as well as unwelding the cryostat. Thus the field homogeneity of the magnets and its influence on the beam has to be thoroughly studied during the project planning phase. Elliptic multipoles, a new type of field expansion for static or quasi-static (here magnetic) two-dimensional fields, are proposed and investigated, which are particular solutions of the potential equation in plane elliptic coordinates obtained by the method of separation. The proper subsets of these particular solutions appropriate for representing static real or complex fields regular within an ellipse are identified. Formulas are given for computing expansion coefficients from given fields. The advantage of this new approach is that the expansion is valid, convergent and accurate in a larger domain, namely in an ellipse circumscribed to the reference circle of the common circular multipoles in polar coordinates. Formulas are derived for calculating the circular multipoles from the elliptical ones. The effectiveness of the approach was tested on many different magnet designs and is illustrated here on the dipole design chosen for the core synchrotron (SIS 100) of the FAIR project as well as on measurement data obtained by rotating coil probes.

23 citations

Journal ArticleDOI
TL;DR: In this paper, the authors search for another coordinate system in the neighborhood of the normal coordinate system so that in the new coordinate system removal of the coupling terms in the equations of motion produces a minimum bound on the relative error introduced in the approximate solution.
Abstract: The method in effect searches for another coordinate system in the neighborhood of the normal coordinate system so that in the new coordinate system removal of the coupling terms in the equations of motion produces a minimum bound on the relative error introduced in the approximate solution

22 citations

Journal ArticleDOI
TL;DR: An algorithm for lossless conversion of data between Cartesian and polar coordinates, when the data is sampled from a 2-D real-valued function expressed as a particular kind of truncated expansion, is proposed.
Abstract: In this paper, we propose an algorithm for lossless conversion of data between Cartesian and polar coordinates, when the data is sampled from a 2-D real-valued function (a mapping: ) expressed as a particular kind of truncated expansion. We use Laguerre functions and the Fourier basis for the polar coordinate expression. Hermite functions are used for the Cartesian coordinate expression. A finite number of coefficients for the truncated expansion specifies the function in each coordinate system. We derive the relationship between the coefficients for the two coordinate systems. Based on this relationship, we propose an algorithm for lossless conversion between the two coordinate systems. Resampling can be used to evaluate a truncated expansion on the complementary coordinate system without computing a new set of coefficients. The resampled data is used to compute the new set of coefficients to avoid the numerical instability associated with direct conversion of the coefficients. In order to apply our algorithm to discrete image data, we propose a method to optimally fit a truncated expression to a given image. We also quantify the error that this filtering process can produce. Finally the algorithm is applied to solve the polar-Cartesian interpolation problem.

22 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20237
202211
202111
202010
201913
201810