  We study a compact family of totally elliptic representations of the
fundamental group of a punctured sphere into $\mathrm{PSL}(2,\mathbb R)$
discovered by Deroin and Tholozan and named after them. We describe a polygonal
model that parametrizes the relative character variety of Deroin--Tholozan
representations in terms of chains of triangles in the hyperbolic plane. We
extract action-angle coordinates from our polygonal model as geometric
quantities associated to chains of triangles. The coordinates give an explicit
isomorphism between the space of representations and the complex projective
space. We prove that they are almost global Darboux coordinates for the Goldman
symplectic form.
