adjusted for easy visual comparison. Also note that the two-arc property does not hold for this example so our reconstructability results do not apply. The accuracies of both the curve reconstruction and area density signature fit suggest that somewhat more general reconstructability results hold. In particular, we speculate that general simple polygons may be reconstructible from g ( s, r ) for fixed r and no derivative information. We have studied the integral area invariant with particular emphasis on the tangent- cone graph-like condition. In particular, we have shown that all TCGL polygons and a C $^{1}$-dense set of C 2 TGL curves are reconstructible using only the integral area invariant for a fixed radius along the boundary and its derivatives. We also showed that TCGL boundaries can be approximated by TCGL polygons, determined what the derivatives represented, and commented on other sets of data sufficient for reconstruction (namely, both T-like and all radii in a neighborhood of 0). These reconstructions are all modulo translations, rotations, and reparametrizations. The arc length parameterization plays a special role here since any two such parame- terizations of a boundary will differ only by a shift and can easily be placed into corre- spondence. The situation becomes more complicated in higher dimensions as boundaries are no longer canonically parameterized by a single variable which is a fundamental as- sumption of our results and methods. It is not immediately obvious how to resolve the issues created by higher dimensions except that it may be possible to modify some of the machinery to work with star convex regions which restore some semblance of canonical representation. 114 4.9 Conclusions