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Pivot points within a 7 point hinge

Please see the attached file for the fully formatted problem(s).

My problem is explained more in my attachment, but briefly, I require some form of equation or graph to calculate where the pivot points within a seven point hinge system need to be in order for the rotating edge to rotate around a origin.

The question is in regard to the hinge system.
I require the hinge, when expanded, to rotate around the origin with 0 tolerance.
THE HINGE
What I know
Through pure trial and error, involving no mathematical formulas or graphs, I have
narrowed down where the pivot points, approximately need to be placed so it rotates
around the origin.
I did this by lengthening and shortening distances and angles between points, until I
narrowed the solution down to be within 2mm tolerance.
As can be seen in the picture below, the plate that will be used to bend the material
moves up/out/in/and down while rotating around the origin. I require this plate to have no
tolerance, i.e. not move away from the origin at all. Therefore I believe there must be
some sort of graphs or equations that will dictate exact positions the pivot points within
the hinge need to be.
Bending Edge
Origin
Bending edge shown here at different stages of rotation

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Solution Summary

The pivot points on a seven-point hinge system are determined.

$2.19