  Suppose a relatively elliptic representation $\rho$ of the fundamental group
of the thrice-punctured sphere $S$ is given. We prove that all projective
structures on $S$ with holonomy $\rho$ and satisfying a tameness condition at
the punctures can be obtained by grafting certain circular triangles. The
specific collection of triangles is determined by a natural framing of $\rho$.
In the process, we show that (on a general surface $\Sigma$ of negative Euler
characteristics) structures satisfying these conditions can be characterized in
terms of their M\"obius completion, and in terms of certain meromorphic
quadratic differentials.
