Pending. There is a logic mis. (i.e. This stage is OK. but, in next stage is NG.) N= odd number is difficult.
N=3 is possible, but rigid body (Reuleaux triangle itself.). My now answer is ↓.
N=3 2R-Virtual Wheel (Reuleaux triangle)
This is typical "Line symmetry linkage" application sample.

Dynamic large radius wheel. N=3 Polygon wheel.
From above "Elementary parts" left figure, we can recognize that the point M_{1} traces on straight line can be explained easily by line symmetry property.
Tip:
In above sample, pink bar edge ratio pp=0.26 is assigned.
Green bar edge ratio is assigned 0.5. (2:1 is, brings easy calculating)
If necessary, please do tuning it.
■ ideal rule
3 Cheese sectors (Black, Blue, Green) are connected by next relation.
(1) Each sector's Top B, B', B''' are belong to the r=R circle 120° sector C'''AC', C'AC'', C''AC''' areas.
(2) There exists Master and slaves relation between B, B', B'''.
The line To Axis A is vertical, such vertex is Master.
in above fig. sample, BA is vertical, so, vertex B is master. B', B''' are slaves.
Master/ slaves relation are dynamically changed.
(3) The line To Axis A, i.e. BA, B'A, B'''A are dependent.
Master and Slave is in Line-symmetry. Slave and Slave has no relations.
ex. BA -- C'A---B'A, BA--C'''A--B'''A are line symmetry.
B'A and B'''A are slaves, and B'A--C''A--B'''A relation is not line symmetry.
Please implementation this, if you can do.