Dividing a circle into areas lines (there are two points on one side and one on the other). So the recurrence can be expressed as which can be easily reduced to using the sums of the first formula_3 natural numbers and the first formula_3 squares, this combines to Finally which yields The lemma asserts that the number of regions is maximal if all "inner" intersections of chords are simple (exactly two chords pass through each point of intersection in the interior). This will be the case if the points on the circle are chosen "in general position". Under this assumption of "generic intersection", the number
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