As you see, the line need to wrap around and around the cylinder. Or you can imagine that the cylinder is rolling and press the line on it. The difficulty is that because the line is long, thus there are more than one round to wrap.
Thanks for any suggestions!
wrap.3dm (530.5 KB)
One thing you’ll need to do is Rebuild your curve. It needs lots of control points for enough “flexibility” to wrap.
Thank you so much for your reply, Ethan!
I will try now.
Sorry, but I don’t think it’s what I want.
In your model the line just runs 360 degrees around the cylinder. But as you see, in my file is line is much longer compared to the cylinder, thus the surface under the line will be much more than the cylinder surface. That means there will be more than one “map”.
If it’s always going to be a helix, as it appears, then you can just define it mathematically, but continuing with my prior def, here we have two curves in the plane which become one on the cylinder, where one begins where the other ends,so it wraps twice. I’m sure you can figure out how to extend this to as many twists as you want.
Smart guy 
Thank you for your help again!
Have a nice weekend.
Wow, Ivelin, you are so nice!! The results is so beautiful!
Thank you for your kind help!
