Spherical Coordinates in n-space

  In a previous blog, Best Position Estimate for a Number of Intersecting Great Circles, I showed how one can find the distance of a point on a sphere from an arc on it. The trick involves finding the point on the arc for which the distance from the point is a minimum. This method can be used to find the angular coordinates of a point, q, on a sphere. The coordinate, φ, is the angular distance to the minimum, a, on the arc in the xy-plane. The second coordinate, θ, is the angular distance from the point a to the point q on the arc in the az-plane. Of course the distance of q from the arc in this case is zero.

   This simple process can be extended to any number of dimensions. In a Euclidean 4-space we have the axes x, y, z and w. The distance of the minimum of b from q on the arc in the az-plane will not necessarily be zero in this 4-space. We need another arc in the bw-plane to get to q on the 4-sphere which yields a third angle. Here's an example of how it works in practice.

  One can use the atan function to check the angles in the example above. The function e_r(α) consists of pairs of sines and cosines as indicated in the blog mentioned above with a third sine-cosine pair for the third angle.

  For points not on the unit 4-sphere the distance of the point q from the origin will not be unity and a check will show that q = r·e_r(α) to the precision of the calculation.