I've been doing some reading on spatial Poisson point processes on my own tonight, and right now having a headache or a brainwarp or I don't know what because I don't get this definition on Wikipedia:
The spherical contact distribution function is defined as:
$H_s ( r ) = 1 − P ( N ( b ( o , r ) ) = 0 )$
where $b(o,r)$ is a ball with radius$ r$ centered at the origin $o$. In other words, spherical contact distribution function is the probability there are no points from the point process located in a hyper-sphere of radius $r$.
In case of homogenous Poisson process, I can see how that $P ( N ( b ( o , r ) ) = 0 )= e^{-\lambda |b(o,r)|}$. The thing I don't get is: why do we subtract this term from $1$?
By construction, isn't $P ( N ( b ( o , r ) ) = 0 )$ alone itself the probability that number of points in a sphere with radius $r$ is $0$. Isn't this exactly what the textual description of definition states?
From that wiki page you shared in your question: "More specifically, a spherical contact distribution function is defined as probability distribution of the radius of a sphere when it first encounters or makes contact with a point in a point process." - Thus it depicts the probability when there is at least one count of point process point(s) within the spacial ball of $b(o,r)$.
Since $\mathbb P(N(b(o,r)=0)$ is the probability of zero points within $b(o,r)$, you do $1-\mathbb P(N(b(o,r)=0)$.
I am recently also learning it. I am reading Liniger 2009, and I recommend it.