The contraints for boundaries of the the different regions

Boundaries are represented as the equation of a 3D plane, intersecting the unit sphere. These intersections are great and small circles. THe representation is in terms of a 4-vector, (x,y,z,c), where (x,y,z) are the components of a 3D normal vector pointing along the normal of the plane into the half-scape inside our boundary, and c is the shift of the plane along the normal from the origin. Thus, c=0 constraints represent great circles. If c<0, the small circle contains more than half of the sky.
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