3-sphere two 3-balls. This construction is analogous to a construction of a 2-sphere, performed by gluing the boundaries of a pair of disks. A disk is a 2-ball, and the boundary of a disk is a circle (a 1-sphere). Let a pair of disks be of the same diameter. Superpose them and glue corresponding points on their boundaries. Again one may think of the third dimension as temperature. Likewise, we may inflate the 2-sphere, moving the pair of disks to become the northern and southern hemispheres. After removing a single point from the 2-sphere, what remains is homeomorphic to the Euclidean
Circle of a sphere satisfy The intersection of the spheres is the set of points satisfying both equations. Subtracting the equations gives In the singular case formula_7, the spheres are concentric. There are two possibilities: if formula_8, the spheres coincide, and the intersection is the entire sphere; if formula_9, the spheres are disjoint and the intersection is empty. When "a" is nonzero, the intersection lies in a vertical plane with this x-coordinate, which may intersect both of the spheres, be tangent to both spheres, or external to both spheres. The result follows from the previous proof for sphere-plane intersections. Circle of a sphere A