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if the zig zag doesnt have sharp corners you could make a round datum curve and offset it twice, once in either direction to represent the maximum width.
create points in roughly the correct postions for the points on the zig zag on the two outisde curves using length ratio.
* you can tweak these positions later by using modify.
The create a curve thru these points.
You may have to create an extruded surface from the original ring and project the resulting zig zag curve on this because you cannot create a swept surface on a 3d curve.
This is hard to explain in text sorry if its confusing.
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