## Mathematical Recreations

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#### Spherical Coordinates

The problem for February 2002 involves spherical coordinates (or spherical-polar coordinates) denoted by (θφ). These angular coordinates on the unit sphere are related to the usual Cartesian coordinates (xyz) as follows:

z  =  cos θ
x  =  sin θ cos φ
y  =  sin θ sin φ

That is, θ is the angle between the z-axis and the line from the center of the sphere at (xyz) = (0,0,0) to the point on the surface of the sphere, and φ is the angle between the x direction and the perpendicular from the point to the z-axis. θ always ranges from zero to pi. φ usually ranges from zero to two pi, but may range from minus pi to pi, for example.

Three-dimensional spherical coordinates include a third dimension r, which is the distance from the center of the coordinate system to the point. For points on the unit sphere, r = 1.

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Thanks,
Steve