Saturday, September 13, 2014

Range of the Spherical Coordinate Angles for 4-space

  What is the range for the three spherical coordinate angles would be in 4-space?  We can do a check by generating a large number of random points on the 4-sphere and see what we get.

The first minimum point, a, and angle, α0, can be any position on a 360° arc in the xy-plane. The remaining angles lie on half arcs and range between ±90°.

Supplemental (Sep 13): One must check to see if the angle found is the actual minimum since an optimal point on the arc may correspond to a maximum or a minimum. To find the minimum one can use the chord of the 4-sphere connecting the reference point (a or b) and the point q which is the linear distance instead of the angular distance. The case where both coordinates of q in the plane of the arc are zero need special attention since the angle is indeterminate but in this case it can be set to zero.

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