polygon’s angle at each vertex. Theorem 4.7.1. Define$_{G}$ ≡ { γ | γ is a C 2 simple closed curve and tangentially graph- like for r = ˆ r } . Suppose that, for r = ˆ r , for all s ∈ [0 , L ] , and for each γ ∈ $_{G}$, we know the first-, second-, and third-order partial derivatives of g$_{γ}$ ( s, r ) . Then the set of recon- structible γ ∈ $_{G}$is C 1 dense in$_{G}$ where reconstructability is modulo reparametrization, translation, and rotation. The first part of the proof shows that the derivative information can be used to obtain the curvature. However, it is not the curvature at the boundary point where the circle is centered but rather the curvature at each of the points where the boundary enters and exits the circle. Although the Euclidean distance to these points is known, the arc length distances are not and can vary from point to point. Thus the sequences of curvatures we obtain also lose the arc length parameterization of our area invariant. The rest of the proof is concerned with finding the arc length distance from the center to the entry and exit points which effectively recovers the curvature for all points. This relies on matching up the unique features of exit angle sequences with each other which in turn relies on the existence of unique maxima and minima in these sequences. While this is not true in general, it can be arranged to be so by a suitable small perturbation of the boundary (which is why our result is one of density rather than for all shapes). This is a theoretical paper about a measure that is useful in applications: we do not pretend that the reconstruction techniques in our proofs are practically useful. In fact, the reconstructions we use to show uniqueness would be seriously disturbed by the noise that any practical application would encounter. We do, however, comment on some possible approaches to reconstruction (section 4.8) using the OrthoMads direct search algorithm [2] to successfully reconstruct shapes which are not predicted by our theory. 80