In this paper, we study quasi-static manipulation of a planar kinematic chain with a fixed base in which each joint is a linearly elastic torsional spring. The shape of this chain when in static equilibrium can be represented as the solution to a discrete-time optimal control problem, with boundary conditions that vary with the position and orientation of the last link. We prove that the set of all solutions to this problem is a smooth three-manifold that can be parameterized by a single chart. Empirical results in simulation show that straight-line paths in this chart are uniformly more likely to be feasible (as a function of distance) than straight-line paths in the space of boundary conditions. These results, which are consistent with an analysis of visibility properties, suggest that the chart we derive is a better choice of space in which to apply a sampling-based algorithm for manipulation planning. We describe such an algorithm and show that it is easy to implement.