Another space which is open for further development is that of reconstruction algo- rithms. This is doubly true since our theoretical reconstructions are unstable and the numerical examples in the present work do not have guaranteed reconstruction. How- ever, even without these guarantees, the numerical examples hint at more expansive reconstructability results. The authors would like to thank David Caraballo for introducing us to this topic as well as Simon Morgan and William Meyerson for initial discussions and work on related topics that are not in this paper. This research was supported in part by National Science Foundation grant DMS-0914809. For completeness, we include a short proof of the fact that knowing g ( s, r ) for all s and r very easily gives us reconstructability. This follows from the fact that knowing the asymptotic behavior of g ( s, r ) as r → 0 for any s gives us κ ( s ). That in turn implies that knowing g ( s, r ) in any neighborhood of the set ( s, r ) ∈ [0 , L ] × { r = 0 } also gives us κ ( s ) and therefore the curve. Theorem 4.11.1. Suppose ∂ Ω is C 2 and there exists ϵ > 0 such that we know g ( s, r ) for all ( s, r ) ∈ [0 , L ) × (0 , ϵ ) . This information is enough to determine the curvature of every point on ∂ Ω . In particular, if γ : [0 , L ) → ∂ Ω is a counterclockwise arclength parameterization of ∂ Ω , then κ ( γ ( s )) = − 3 π lim$_{r}$$_{→}$$_{0}$ ∂ ∂r g ( s,r ) πr 2 . Proof. Fix s ∈ [0 , L ). If the curvature of γ at s is positive, we consider what happens if 115 4.11 Appendix: Easy Reconstructability 4.10 Acknowledgments