  We present a way to build hyperbolic spheres with conical singularities by
gluing together simple building blocks. Our construction provides good control
over the holonomy of the resulting hyperbolic cone sphere. In particular, it
can be used to realize any Deroin-Tholozan (DT) representation as the holonomy
of a hyperbolic cone sphere. Our construction is inspired by the correspondence
between DT representations and chains of triangles in the hyperbolic plane. It
gives a geometric interpretation of certain action-angle coordinates on the
space of DT representations, which come from this correspondence.
