How to project a sketch onto multiple surfaces at once?

Hi all,

In the image below, I would like to project the sketch onto the revolve. I would like to project onto all of its surfaces at one time, if possible. Please note that the sketch overlaps between different surfaces. Preferably, I'd like to do this such that I could even extrude/offset the sketch from the surface.

Anybody have any suggestions? Thank you in advance!

Thank you,

Jeff Houston