For a cylinder with equation $r^{2} = y^{2} + z^{2}$, is there a test which could tell me if a closed curve (a connected set of finite points on the surface of the cylinder, for which I know their coordinates in cartesian and cylindrical coordinates) wraps circumferentially around the cylinder like a ring on a finger (1), or lays on the cylinder like an anchor tattoo on a sailor's arm (2)?
Project the points down to the $xy$-plane (ignore the $z$ coordinate). Start at one of the points. March along the sequence of points incrementing the angle from the origin as you go. You will end up with an integral multiple of $2\pi$ when you return to the starting point. The integer multiple will tell you how many times you wound around (and in which direction). The tattooed anchor will give $0$.