Random Geometry on the Sphere and its Applications

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1. For a graph G with N edges, put its vertices on the d-dimensional unit sphere. Let D denote the minimum spherical distance between a pair of points that correspond to a pair of adjacent vertices in G. Then, it was proved that the distribution of ND^d tends to exponential distribution with mean dB(1/2,d/2) as N tends to infinity, where B(p,q) denotes the beta function.2. Let F={C_1,C_2,…,C_N} be a family of caps on the two dimensional unit sphere. A cap C_i is called extremal if the centers of those caps that intersect C_i are all contained in the same side of a great circle passing through the center of C_i. A cap that is smaller than a hemisphere is called proper. It was proved that if F has no extremal cap then the intersection graph G(F) of F is connected. If furthermore, all caps in F are proper then G(F) is 2-connected. For higher dimensional sphere, the similar result never holds. Applying this the following asymptotic result was proved. Now, let F denote a family of N random caps all of the same size (4πc/N)log N. If c>1/2, then the probability that G(F) is 2-connected tends to 1 as N tends to infinity. If c<1/4, then the probability that G(F) is connected tends to 0 as N tends to infinity.3. Let AOB be a triangle in the 3-space with angle ∠AOB=ω. When we look at this angle from a viewpoint P, this angle looks as though the angle of the orthogonal projection of AOB on a plane perpendicular to the line PO. And its size changes according to the location of the viewpoint P. If P is a random point on a unit sphere centered at O, then the 'visual' size of the angle ∠AOB is called the random visual size and denoted by Θ(ω). By a joint study with Yoich Maeda (Tokai univ.), we proved that the expected value of Θ(ω) is equal to ω, and derived a formula to calculate the variance of Θ(ω).