For a more tortured line

Something like that

Project your curve on the ring
Extrude to Z
take a thickness Shell /middle center line of this curve out of the ring
Move it a little inside the ring
Trim from the ring
Network the hole "intersection"
Show Points
and move the line of middle points as above! :)
Join





Maybe like Chipp Walter tuto a "double Trim + Blend" will be necessary following the material used in render
for have a nice junction!