distance between two points on a sphere in spherical coordinates
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The spherical distance between two points  and on a sphere is the distance of the shortest path along the surface of the sphere (paths that cut through the interior of the sphere are not allowed) from  to , which always lies along a great circle.
For points  and  on the unit sphere, the spherical distance is given by

where  denotes a dot product.
For points  and  on the unit sphere, the spherical distance is given by

where  denotes a dot product.
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