This is Lemma 7.17.3 in Beardon:
Lemma 7.17.3 Let $L$ and $L'$ be geodesics in the hyperbolic plane. Then the inversive product $(L,L')$ is $\cosh{\rho(L,L')},1,\cos{\phi}$ according as $L,L'$ are disjoint, parallel or intersecting at an angle $\phi$ where $0\leq\phi\leq\pi/2$ where $\rho(L,L')$ denoted as the length of common perpendicular of $L,L'$.
Beardon defined in higher dimension, but here I will just restrict it to the plane. We denote a circle $S(a,r)$ as $|x|^2-2(x\cdot a)+|a|^2-r^2=0$ where $x=(x_1,x_2),a=(a_1,a_2)$ and its coefficient vector as $(1,a_1,a_2,|a|^2-r^2)$. Here is the definition of inversive product:
Definition 3.2.2 Let $\Sigma,\Sigma'$ have coefficient vectors $(a_0,\cdots,a_{n+1})$ and $(b_0,\cdots,b_{n+1})$ respectively. The inversive product $(\Sigma,\Sigma')$ is $$(\Sigma,\Sigma')=\frac{|2(a\cdot b)-a_0b_{n+1}-a_{n+1}b_0|}{2||a|^2-a_0a_{n+1}|^{1/2}||b|^2-b_0b_{n+1}|^{1/2}}$$. We can write inversive product more concisely as $$(\Sigma,\Sigma')=\frac{|q(a,b)|}{|q(a,a)|^{1/2}|q(b,b)|^{1/2}}$$ where $q(x,y)= 2(x_1y_1+\cdots+x_ny_n)-(x_0y_{n+1}+x_{n+1}y_0)$.
My first question: without further application, can we have any information why inversive product is interesting or the reason it should be defined in this way? It seems mysterious for me since it looks like this definition comes from nowhere due to my lacked knowledge.
One can show inversive product is invariant under isometries. Some remarks that makes it very convenient for hyperbolic computation. For example, it is quite useful that if $\Sigma=S(a,r)$ and $\Sigma'=S(b,t)$ intersects, then $(\Sigma,\Sigma')=|\frac{r^2+t^2-|a-b|^2}{2rt}|=\cos{\gamma}$ where $\gamma$ is the angle of intersection. If $\Sigma$ is a vertical line $x=t$ and $\Sigma'=S(a,r)$, then $(\Sigma,\Sigma')=d_{euc}(a,t)/r$.
Second question: I want to verify Lemma 7.17.3 when $L,L'$ are ultraparallel and I'm confused about a part of the proof.
In the proof, he wrote:
By the usual invariance arguments we need only consider the cases (i) $L,L'$ are in $H^2$ and are given by $|z|=r,|z|=R$; (ii) $L,L'$ are in $H^2$ and are given by $x=0,x=x_1$; (iii) $L,L'$ are Euclidean diameters of the disk model $\Delta$.
However, why could we consider (i) by the invariance arguments? It makes sense for me to consider the case $x=0, |z-a|=R$ for some $a\in\mathbb{R}$. But I'm a little confused about $(i)$.
The classical way: up to apply isometries, we can assume $L$ is the vertical axis with endpoints $0,\infty$ and $L'=S(a,r)$. The existence of common perpendicular and angle of parallelism tells us $\cosh{\rho(L,L')}=1/\sin{\theta}=a/r$. Since $L$ has coefficient vector $(0,1,0,0)$ and $L'$ as $(1,a,0,a^2-r^2)$, by computation we have $(\Sigma,\Sigma')=a/r$ and the claim holds true.