Drawing animation on a path

We need to make kind of a path animation out of a string (curved lines) drawing. The designer provided me with this: http://swfaddress.arb-tb.ro/lfi001string.jpg

We need a revealing (growing) type of animation that generates from the right, follows every corner and ends up in the left with that giant piece of “mop”.

Is there a way to do this professionally? Tried the old frame by frame masking, but it looks odd. Shouldn’t be some AS3 libraries that just work with paths or that draw paths? How would you guys do it? Thanks.