constraintscircle packing to covering on a sphere with antipodal

varies from the maximum antipodal packingradius to the minimum antipodal covering radius. These cases exhibit sequences ofsymmetry- and edge-number transitions in the contact polyhedra, which, in general,differfromthoseoftheunconstrainedproblemwherecentrosymmetryisnotenforced.New minimum antipodal covering arrangements are conjectured for 4, 5 and 7 pairs:they are a hexagonal bipyramid, a