How do you calculate distance between two cylindrical coordinates?

Question

I can't figure out how to find the distance between these two points, expressed with cylindrical coordinates:
$P1 = (9.5 m, 1.00531 rad, 18.2 m)$
$P2 = (9.75 m, 5.27788 rad, 18.2 m)$

What is the precise method to find the distance between these two points? What is the formula? Thanks.

Answer 1

You can use the law of cosines for the $r,\phi$ plane, then combine that with the $z$ difference (which you don't have here).
$$d^2=r_1^2+r_2^2-2r_1r_2 \cos(\phi_1-\phi_2) + (z_1-z_2)^2$$

Answer 2

The two points have cartesian coordinates $(x_i,y_i,z_i)$ given by $$x_i=r_i\cos\phi_i,\quad y_i=r_i\sin\phi_i, \quad z_i=z_i\ .$$ Their distance then is $$d=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2+(z_1-z_2)^2}\ .$$