Random Geometry on the Sphere and its Applications

1.对于一个有N条边的图G，将它的顶点放在d维单位球面上。设D表示G中一对相邻顶点对应的一对点之间的最小球面距离。然后证明了当N趋于无穷大时，ND^d的分布趋于平均dB（1/2，d/2）的指数分布，其中B（p，q）表示beta函数.设F={C_1，C_2，.，C_N}是二维单位球面上的帽族.如果与C_i相交的帽的中心都包含在通过C_i中心的大圆的同一侧，则称帽C_i为极值。比半球小的帽被称为适当的。证明了如果F没有极帽，则F的交图G（F）是连通的。如果F中的所有帽都是真帽，则G（F）是2-连通的。对于更高维的球体，类似的结果永远不成立。应用这一点，证明了下面的渐近结果。现在，让F表示一个N个随机帽族，所有帽的大小都相同（4πc/N）log N。如果c>1/2，则G（F）是2-连通的概率随着N趋于无穷而趋于1。如果c<1/4，则G（F）连通的概率随着N趋于无穷而趋于0.设AOB是三维空间中的一个三角形，其角为Δ AOB=ω。当我们从视点P看这个角度时，这个角度看起来好像是AOB在垂直于线PO的平面上的正交投影的角度。如果P是以O为中心的单位球面上的一个随机点，那么角度θ AOB的“视觉”大小称为随机视觉大小，并由Θ（ω）表示。通过与前田洋一（东海大学）的联合研究，证明了Θ（ω）的期望值等于ω，并导出了计算Θ（ω）方差的公式。

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).