How to attach a sub-curve to a rotating curve

I have three curves... a base curve (ellipse like), a sub-curve extracted from the base curve, and a polygon.

The polygon rotates inside the base curve.

I need to have the sub-curve "welded" to the polygon curve at the polygon apex points such that the sub-curve rotates with the polygon. 

What I'm trying to do is replace the straight curve segment of the polygon with a different curve (the extracted sub-curve.)

Anyone have some hints on how I can do that?