where H m is the m -dimensional Hausdorff measure and α$_{m}$ is the volume of the unit ball in R m [33]. In the current context, the 2-dimensional density of Ω at γ ( s ) is simply While we can evaluate this for all s ∈ [0 , L ), just knowing the density at every point along the boundary is generally insufficient to reconstruct the original shape. If γ $^{′}$( s ) exists, then Area(Ω ∩ D ( γ ( s ) , r )) is approximated arbitrarily well for sufficiently small r by replacing ∂ Ω with its tangent line (which gives us an area of exactly πr 2 2 ). Hence, we have Θ $^{2}$(Ω , γ ( s )) = 1 2 at any point where γ is differentiable. That is, just knowing Θ 2 (i.e., the limit) is insufficient to distinguish any two shapes with C 1 boundary. Contrast this with the situation where we know Area(Ω ∩ D ( γ ( s ) , r )) for every s ∈ [0 , L ) and r > 0 (i.e., we have all of the values needed to compute the limit as well). This added information is sufficient to uniquely identify C 2 curves by recovering their curvature at every point (see Appendix 4.11). One natural question to ask (and the focus of the present work) is whether failing to pass to the limit (i.e., using some fixed radius r instead of the limit or all r > 0) and collecting the values for all points along the boundary preserves enough information to reconstruct the original shape. That is, can a nonasymptotic density (perhaps along with information about its derivatives) be used as a signature for shapes? Definition 4.2.1. In the current context, the integral area invariant [30] is denoted by g : [0 , L ) × R + → R + and given by 4.2.1 Definitions Θ $^{2}$(Ω , γ ( s )) = lim r ↓ 0 Area(Ω ∩ D ( γ ( s ) , r )) πr 2 . g ( s, r ) = ∫ D ( γ ( s ) ,r ) ∩ Ω dx = Area(Ω ∩ D ( γ ( s ) , r )) . 82