I am looking for references describing the structure of the set of immersions of the disk $D^2$ in the Euclidean plane $\mathbb{R}^2$ or in the hyperbolic plane $\mathbb{H}^2$ such that the boundary is piecewise geodesic (up to reparametrisation and isometries of $\mathbb{R}^2$ or $\mathbb{H}^2$). In other words, I am looking for a description of the set of immersed polygons either Euclidean or hyperbolic, up to isometries.

The work Kapovich-Millson in 1995 seems to describe only polygonal curves, without the requirement that these curves can be filled by immersed disks. Is it possible to deduce from this an explicit description of the set of immersed polygons? Is this space has already been studied somewhere?