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TopoZone can give you coordinates for a number of U.S. place names free. From the USGS, so biased towards natural/national landmarks. The topozone interface has changed, so be sure to select NAD83 coordinate system, as it is no longer the default. Epodunk will give you coordinates for a number of U.S. place names free. Seems to be biased ...
Below the complex spherical harmonics are represented on 2D plots with the azimuthal angle, , on the horizontal axis and the polar angle, , on the vertical axis.The saturation of the color at any point represents the magnitude of the spherical harmonic and the hue represents the phase.
Once the radius is fixed, the three coordinates (r, θ, φ), known as a 3-tuple, provide a coordinate system on a sphere, typically called the spherical polar coordinates. The plane passing through the origin and perpendicular to the polar axis (where the polar angle is a right angle) is called the reference plane (sometimes fundamental plane).
In the cylindrical coordinate system, a z-coordinate with the same meaning as in Cartesian coordinates is added to the r and θ polar coordinates giving a triple (r, θ, z). [8] Spherical coordinates take this a step further by converting the pair of cylindrical coordinates (r, z) to polar coordinates (ρ, φ) giving a triple (ρ, θ, φ). [9]
Consider the projection centered at S = (0, 0, −1) on the unit sphere, which is the set of points (x, y, z) in three-dimensional space R 3 such that x 2 + y 2 + z 2 = 1. In Cartesian coordinates (x, y, z) on the sphere and (X, Y) on the plane, the projection and its inverse are then described by
Consider the function R 3 → R 4 given by (x, y, z) ↦ (xy, xz, y 2 − z 2, 2yz). This map restricts to a map whose domain is S 2 and, since each component is a homogeneous polynomial of even degree, it takes the same values in R 4 on each of any two antipodal points on S 2. This yields a map P 2 (R) → R 4. Moreover, this map is an embedding.
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The three surfaces intersect at the point P with those coordinates (shown as a black sphere); the Cartesian coordinates of P are roughly (1.0, −1.732, 1.0). Cylindrical coordinate surfaces. The three orthogonal components, ρ (green), φ (red), and z (blue), each increasing at a constant rate. The point is at the intersection between the ...