Moduli space for Quadrilaterals and Polygons

How the spaces of all quadrilaterals look? What if we fix the side lengths?
The analogous question for triangle is not that interesting, if we fix the side length up to plane isometries there are only two triangle with given side lengths that are obtained by intersection of two circles drawn on a base side, so they are also congruent one obtained from another by reflection through the base.
But we can deform a quadrilateral in many ways. If we draw circles on a base side with radii of two other sides, we see that there are portions of the circles that give rise a points that can be arranged to be at a distance equal to the fourth side, thus the parameter space is closely related to the circles. In fact, depending on the side lengths different topologies can occur.

If we have the equality for side length , we get a degenerate quadrilateral that cannot be deformed further, so the moduli space is just a point.
There are choices where the space is just a circle (topologically). There are cases where the space is disjoint union of two circle — for instance take the triangle, and deform near a vertex slightly to create a four sides with a very very small side, then we can see that the moduli space for triangles consisting of two points now deforms to two small disjoint circles.
There are cases where the moduli space is two circles that are wedged at a single point. This occur when the sides lengths of the previous cases are deformed so that the two circles intersect at a point corresponding to a degenerate quadrilateral.
In the special cases where we have , the space is two circle that are wedge together at two distinct points.
Finally when all the sides are equal the space is a union of three circles where every pair of circles are wedged at some point.

If the sides lengths don’t allow for degenerate quadrilaterals, we get either a circle or disjoint union of circle both which are smooth manifolds. In general, degenerate points give the singular points where smooth parts are attached. The space is disconnected only in cases like those close to the triangle where there are three very long sides compared to rest of the sides. This is true even for moduli space of polygons. The spaces here are also non-singular smooth manifolds provided we don’t have degenerate polygons and the spaces look like toruses . In the near triangle cases, we it’s easy to see that we can union of two dimensional toruses. The general picture is that that the sides lengths satisfy some triangle inequalities and points satisfying these inequalities is a simplex, and there are points of the simplex corresponding to the degenerate polygons near which the topology of the moduli space changes.

http://www.math.umd.edu/~millson/papers/space.pdf
https://projecteuclid.org/journals/journal-of-differential-geometry/volume-42/issue-1/On-the-moduli-space-of-polygons-in-the-Euclidean-plane/10.4310/jdg/1214457034.full