What is the parametric equation for a helix on a vertical, circular cylinder?

[dlacombe13]

Homework Statement

Match the parametric equations with the graphs.
In this case, I am stuck on this equation:
x = cos t
y = sin t
z = 1/(1+t^2)

Homework Equations

The Attempt at a Solution

So far I have:
x^2 + y^2 = cos ^2 t + sin ^2 t = 1
I know this is a circle in the xy-plane, and thus this yields a vertical, circular cylinder. I understand that there will be some form of helix going counter-clockwise around the cylinder. However, I do not understand how exactly to "graph" this using the parameter t. I know what the graph is and looks like, but I can't understand why it looks the way it does. I am having a hard time grasping this section in total.

 

[andrewkirk]

Think of an infinitely long, infinitely thin slinky spring whose top is in the plane $z=1$, centred on the point (0,0,1). As you follow the spring down, in the negative direction of the $z$ axis, the coils get closer and closer to one another so that it approaches but never quite touches the plane $z=0$.