4.2.2). These simplifying assumptions allow us to explicitly analyze what happens when we move along the boundary and to work locally without worrying about global effects. We show that the tangent-cone graph-like property can be preserved when approx- imating a shape with a polygon (section 4.3) and discuss what the derivatives of these nonasymptotic densities represent (section 4.4) and show that all tangentially graph-like boundaries can be reconstructed (modulo translations and rotations) given sufficient in- formation about the nonasymptotic density and its derivatives (section 4.5 and appendix 4.11). The main contribution of this paper is to show (under our tangent-cone graph-like condition) that all polygons (theorem 4.6.1 in section 4.6) and a C $^{1}$-dense set of C 2 boundaries (theorem 4.7.1 in section 4.7) are reconstructible (modulo translations and rotations). We briefly discuss and sketch the proofs of these two theorems. Theorem 4.6.1. For a polygon Ω which is tangent-cone graph-like with radius r , suppose that we have the integral area invariant g ( s, r ) where s is parameterized by arc length. Suppose that for all s we know g ( s, r ) and its first derivatives with respect to r (disk radius) and s (position along the boundary). This information is sufficient to completely determine Ω up to translation and rotation; that is, we can recover the side lengths and angles of Ω . The proof of this theorem uses the discontinuities in the s derivative to determine the locations of vertices (and thus the side lengths between them). We combine the r derivative and the one-sided s derivative information when centered on a vertex to recover the angles at which the polygon enters and exits the circle (which might not be the polygon vertex angle if the circle contains another vertex). Doing this with the other one-sided s derivative gives the same thing but using the orientation determined by the other polygon side incident to the vertex. The combination of these yields the 79