Area and perimeter of pseudo-hyperbolic ball We have that the pseudo-hyperbolic metric in the open unit disk $\mathbb D$ is defined by
$$ \rho(z,w) = |\phi_w(z)|,  \qquad \phi_w(z) = \frac{w - z}{1 - \overline w z}$$
where $z,w \in \mathbb D.$ It is also know that $\phi_w(z)$ is an automorphism of the unit disk.
Then for $r$, $0 \lt r \lt 1$ and  $\alpha \in \mathbb D$, the set
$$ P(\alpha,r) = \{ z \in \mathbb D : \rho(z,\alpha) \lt r \}$$
is the pseudo-hyperbolic ball with center $\alpha$ and radius $r.$
I need the formulas of area and perimeter of pseudo-hyperbolic ball, I know that the pseudo-hyperbolic ball is a Euclidean ball, So I think that the formulas for the area and perimeter of the pseudo-hyperbolic ball are the same as the formulas for the area and perimeter of the Euclidean ball.
I am right?
 A: The properties of a hyperbolic ball are different to a Euclidean ball: for example, in your model, the ball $P(0, r)$ has a ‘hyperbolic perimeter’ with respect to $ρ$ which approaches infinity as $r → 1$. (Of course, its normal perimeter with respect to the Euclidean metric is still $2πr$.)
This is because the space near the edge of the hyperbolic disk is “squished” — a small distance in $ℂ$ near the edge corresponds to a large distance according to $ρ$. To calculate the scaling factor, imagine an infinitesimal line segment connecting $z ∈ $ to $z + ε ∈ $, for a small $ε ∈ ℂ$. Its length is
$$
ρ(z, z + ε) = |ϕ_z(z + ε)| = \left|\frac{-ε}{1-z(z + δ)}\right|
= \frac{|ε|}{1 - r^2} + (ε^2)
$$
where $r = |z|$. So the scaling factor is $(1 - r^2)^{-1}$.
To find the hyper-perimeter of a ball $P(α, r)$, we need only consider $α = 0$. Balls at other locations $P(α', r)$ have the same hyper-perimeter, since they can be translated isometrically by $ϕ$ to $P(0, r)$.
Each line element on the perimeter of $P(0, r)$ gets scaled by the same amount, so the hyper-perimeter is $2πr/(1 - r^2)$.
You can then compute the area by integration:
$$
A(0, r) = \int_0^r \frac{2πs}{1 - s^2} ds = -\log \sqrt{1 - r^2} > 0
$$
