In our paper “Probability theory of random polygons from the quaternionic viewpoint”, Jason Cantarella, Tetsuo Deguchi, and I defined a Riemannian metric on the space of closed polygons of fixed total length either in the plane or in 3-space. This measure is pushed forward from Haar measure on the Grassmannian of (real or complex) 2-planes in (real or complex) \(n\)-space, so it is highly symmetric.

In particular, we can explicitly find geodesics in the space of closed polygons. The animation above shows a particular geodesic in the space of closed 12-gons of fixed total length 2.