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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).

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    $\begingroup$ Hi and welcome. It is good that you use some typesetting. On the site MathJax which is similar to LaTeX math syntax is preferred over HTML. There is a tutorial here somewhere on the site. $\endgroup$ – mathreadler Aug 30 '17 at 9:57
  • $\begingroup$ Here it is math.meta.stackexchange.com/questions/5020/… $\endgroup$ – mathreadler Aug 30 '17 at 10:01

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