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Preliminaries
Published in Patrick Knupp, Stanly Steinberg, Fundamentals of Grid Generation, 2020
Patrick Knupp, Stanly Steinberg
In two dimensions, there are many transformations that are useful for generating boundary-conforming coordinate systems; their use predates grid generation. For example, polar coordinates are useful in problems with circular symmetry. Similarly, in three dimensions, spherical coordinates are useful in problems with spherical symmetry. These transformations usually entail analytic formulas and are restricted to fixed physical domains with fixed boundary parameterizations. The most elementary of such coordinate systems in the plane is, of course, the Cartesian system, () x(ξ,η)=ξ,y(ξ,η)=η,
The Miñano Design Method Using Poisson Brackets
Published in Julio Chaves, Introduction to Nonimaging Optics, 2017
The device being designed will have a circular symmetry. Because i3-lines are vector flux lines, the final device will be limited by two of these lines converted to mirrors (with circular symmetry). The points where these two lines cross the entrance aperture (i3 = 0) will then define the entrance aperture of the final device, and the points where these two lines cross the exit aperture will define the exit aperture of the final device. Because each one of these lines crosses the entrance aperture at ρ = i1, and the exit aperture at ρ = ρr, the ratio between the diameters for the entrance and exit aperture will be i1/ρr. The geometrical concentration for the concentrator with circular symmetry will then be Cg = (i1/ρr)2. From expression (13.134), we then obtain
Multi-mode Beams and Apertures
Published in J. C. G. Lesurf, Millimetre-wave Optics, Devices and Systems, 2017
Most optical systems employ elements which are circular and the beam axis is generally arranged to pass through their centres. We can therefore regard such an element as being placed within a circular aperture through which the beam must pass. Given the circular symmetry which arises in most cases it is convenient to make use of a cylindrical coordinate system.
Cylindrically symmetric rotating crystals observed in crystallization process of InSiO film
Published in Science and Technology of Advanced Materials: Methods, 2023
Bo Da, Long Cheng, Xun Liu, Kunji Shigeto, Kazuhito Tsukagoshi, Toshihide Nabatame, Zejun Ding, Yang Sun, Jin Hu, Jiangwei Liu, Daiming Tang, Han Zhang, Zhaoshun Gao, Hongxuan Guo, Hideki Yoshikawa, Shigeo Tanuma
In the EBSD-angle map, different colors represent the relative out-of-plane direction between the local orientation and the average crystal direction of the entire crystallization region. The EBSD-angle map shows that the angle of the central region of the rotating crystal island nearly overlaps with the average crystal direction, which implies two-dimensional spherulitic crystal growth. Furthermore, the relative misorientation angle changes from 0° to 14° from the center to the edge, forming a roughly circular-symmetric distribution. This circular symmetry implies that the relative misorientation angle increases at a constant rate with the distance from the central region.