Random Geometry on the Sphere and its Applications

球体上的随机几何及其应用

基本信息

批准号：
13640126
负责人：
MAEHARA Hiroshi
金额：
$ 1.34万
依托单位：
University of the Ryukyus
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2001
资助国家：
日本
起止时间：
2001 至 2002
项目状态：
已结题

项目摘要

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).

1。对于具有n个边缘的图G，将其顶点放在D维单元球体上。令D表示与G中的一对相邻顶点相对应的一对点之间的最小球体距离。然后，证明ND^d的分布倾向于用平均DB（1/2，d/2）指数分布，因为N趋于无限，其中B（p，q）表示beta函数。2。令F = {C_1，C_2，…，C_N}成为二维单位球体上的帽子家族。如果与C_I相交的CAP的中心都包含在通过C_I中心的大圆圈的同一侧，则称为cap c_i。小于半球的盖称为正确。事实证明，如果F没有极端盖，则连接F的相交图G（F）。如果此外，F正确的F中的所有帽子都会正确，则G（f）是2。对于更高的维球，类似的结果永远不会成立。应用这一点证明了以下不对称结果。现在，让f表示一个n随机盖的家族所有相同大小（4πc/n）log n的家族。如果c> 1/2，则g（f）与n趋向于无穷大的概率趋于1。如果c <1/4，则连接g（f）的概率趋向于0，因为n趋于无穷大。3。让AOB为3个空间中的三角形，其角度∂AOB=ω。当我们从视点p看这个角度时，这个角度看起来好像AOB在垂直于线PO的平面上的正交投影的角度。并且其大小根据视点P的位置而变化。如果P是以O为中心的单位球上的随机点，则角度∂AOB的“视觉”大小称为随机视觉大小，并用θ（ω）表示。通过与Yoich Maeda（Tokai Univ。）的联合研究，我们证明了θ（ω）的期望值等于ω，并得出了计算θ（ω）方差的公式。