How to roll a planar curve along a curved developable surface as if it were a development?


I have created a curved surface by sweeping an arc along a line. Obviously this is a developable surface.

Now I would like to take a planar curve (ellipse) and ‘roll’ it along the curved surface so that the planar ellipse is no longer planar. Almost as if this planar curve was actually a planar surface and is used as a development and rolled into the shape of the curved surface. How do I do this?

Thanks for your help

FYI in case you want to know why I’m trying to do this it’s because I’m modelling a chair similar to the one in the jpeg I attached. I have already tried positioning this planar ellipse in the ZX plane and extruding through the curved surface then delete intersecting surfaces. Projecting the curve onto the surface and trimming would produce the same result. It works however it’s a little difficult to easily model the exact shape I want on the right/left side of the chair back where it’s more curved. I think it will be a lot easier to model the exact shape I want if the curve was used as a development instead of extruding/projecting.

Grasshopper Question Rhino.3dm (210.7 KB)

Something like this? (11.7 KB)

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Yes thank you!!