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
Area of a circle value of formula_72 at x. A spherical circle is the set of points a geodesic distance "R" from the zenith point z. Equivalently, with a fixed embedding into formula_73, the spherical circle of radius formula_74 centered at z is the set of x in formula_64 such that formula_76. We can also measure the area of the spherical disk enclosed within a spherical circle, using the intrinsic surface area measure on the sphere. The area of the disk of radius "R" is then given by More generally, if a sphere formula_78 has radius of curvature formula_79, then the area of the