Probabilities that Random Spherical Caps are in Contact

Consider four random spherical caps of common radius on the unit sphere, and assume that their centers, p1, p2, p3, p4, are generated independently and uniformly. Let G be a graph made of four vertices v1, v2, v3, v4, for which vertices vi, and vj are connected by an edge if and only if spherical caps with centers pi, pj are in-contact. We study the following two events: EI that G is composed of a connected triangle of three vertices and an isolated vertex; EII that G is composed of a connected line-segment of four vertices. Since both events occur with zero probability, it is impossible to define the ratio P(EI) : P(EII) by the usual manner. However, by introducing a concept of ε-contactness and later letting ε be arbitrarily small, we can invest a legitimacy to the ratio in an asymptotic sense. On this foundation an exact expression for the ratio will be derived and then it will be evaluated numerically for various radii of speherical caps