Parametric formulas for moving Tool through a Plate

Hi to all

In a analyses software i want write a formula for a tool that move though the plate.

I already use below formula for move the tool on a circle path.

it is a cylindrical coordination.

x : r*sin(\omega *t)

y: r*cos(\omega *t)

I showed path of tool on plate:

http://i813.photobucket.com/albums/z...Untitled-1.png

i what have a formula for move the tool on all surface, step by step like path that showed on the picture.

I think it must be in cylindrical coordination.

I am wait for your answer.

My english is not good . please tell me if you don't understand.

Thank you