Cylindrical coordinates infinitesimal distance is given by:
$d^2s = d^2z + d^2r + r^2 d^2θ$
And formula for squared distance between two points $(θ_1, r_1, z_1)$ and $(θ_2, r_2, z_2)$ is:
$distance^2 = (z_2 - z_1)^2 + r_1^2 + r_2^2 - 2 r_1 r_2 \cos(θ_2 - θ_1)$
I need to find distance between two points $(θ_1, r_1, z_1)$ and $(θ_2, r_2, z_2)$ when infinitesimal distance is:
$d^2s = d^2z + f^2(z) d^2r + f^2(z) r^2 d^2θ$
Essentially, $f(z)$ used for scaling to capture intuition that at certain values of $z$, distance depends only on $z$ ($f(z) = 0$ at this points, but it is positive otherwise).