These equations are used to convert from spherical coordinates to rectangular coordinates. Rectangular coordinates \((x,y,z)\), cylindrical coordinates \((r,θ,z),\) and spherical coordinates \((ρ,θ,φ)\) of a point are related as follows:Ĭonvert from spherical coordinates to rectangular coordinates HOWTO: Converting among Spherical, Cylindrical, and Rectangular Coordinates The radius of the circles increases as \(z\) increases. Either or is used to refer to the radial coordinate and either or to the azimuthal coordinates. Unfortunately, there are a number of different notations used for the other two coordinates. We also find that the finite resolution reflection coefficient achieves its minimum value at a σmaxm > σmaxc Numerical experiments validate the analysis.\): The traces in planes parallel to the \(xy\)-plane are circles. Cylindrical coordinates are a generalization of two-dimensional polar coordinates to three dimensions by superposing a height () axis. For fixed discretization parameters, layer width, and a quadratic loss function, we find the numerical reflection produced by the discrete layer is accurately predicted by the infinite resolution reflection coefficient for σmax ∈ ,where σmax is the maximum value of the absorption parameter in the layer. We also find that the finite resolution reflection coefficient achieves its minimum value at a σmaxm > σmaxc Numerical experiments validate the analysis.ĪB - We present an analysis of the perfectly matched layer in cylindrical coordinates discretized with a staggered second-order accurate finite difference time domain method. N2 - We present an analysis of the perfectly matched layer in cylindrical coordinates discretized with a staggered second-order accurate finite difference time domain method. The author is with the Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ 07102 USA (e-mail: Digital Object Identifier 10.1109/TAP.2003.813626 This work was supported in part by Air Force Office of Scientific Research, Air Force Materials Command, USAF under AFOSR Grant F4-0031. The tangent function of an angle in a triangle is equal to the opposite side divided by the adjacent side. Therefore, we have the relationship: r 2 x 2 y 2 r x 2 y 2 To find the angle, we use the inverse tangent function. It is convenient to use cylindrical coordinates where azimuthal symmetry is obtained. Manuscript received Octorevised May 20, 2002. The x and y coordinates form the legs of the triangle and r forms the hypotenuse. Maxwells equations with the use of finite difference method. T1 - An analytical study of the discrete perfectly matched layer for the time-domain Maxwell equations in cylindrical coordinates
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