The center points
of the equal circles, projected to the sphere surface, constitute the set
of points whose minimum distance is maximal. There are as many minimal
edges in the convex hull of this set as their are circle contacts. A few
circles may be loose and not touching their nearest neighbors. But
the vast majority are rigid. If m is the number of rigid circles
there will be at least 2m-2 minimal edges in the convex hull. This
provides enough equations to solve for the coordinates of the non-loose
points. The equations to be solved, coming from the distance formula,
are polynomial. It follows (when a solution of the equations exists)
that the coordinates of the points (given suitable placement of the polyhedron
i.e. convex hull) can all be taken as algebraic numbers. Since *u*
= cosine( central angle to minimal edge ) is just the dot product of unit
vectors to the endpoints of the edge, it is an algebraic expression in
the coordinates of those endpoints. Hence *u* must be algebraic
also.
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