Taken together, the last two claims give us the curvature as a function of arclength. This determines γ up to translations and rotations. ■ 4.8 Numerical experiments In this section, we consider a numerical curve reconstruction for the situation in which g ( s, r ) is known for a given radius r but no derivative information is available. This reconstruction is more strict than the scenarios of sections 4.5–4.7. Our motivation is to explore whether any γ can be uniquely and practically reconstructed with this limited information. We consider γ$_{a}$ (¯ s ) ∈ P $^{N}$, the set of simple polygons of N ordered vertices { ( x$_{1}$, y$_{1}$ ) , . . . , ( x$_{N}$, y$_{N}$ ) } parameterized by the set { ¯ s$_{k}$ } N k =1 with ¯ s$_{k}$ = k/N as x$_{k}$ = m − 1 ∑ j =0 a$_{1}$$_{,j}$ cos(2 πj ¯ s$_{k}$/N ) + a$_{2}$$_{,j}$ sin(2 πj ¯ s$_{k}$/N ) , y$_{k}$ = m − 1 ∑ j =0 a$_{3}$$_{,j}$ cos(2 πj ¯ s$_{k}$/N ) + a$_{4}$$_{,j}$ sin(2 πj ¯ s$_{k}$/N ) , (4.1) for some coefficients a$_{i,j}$ ∈ R . In this way, the polygon γ is a discrete approximation of a C ∞ curve. The sides of γ$_{a}$ (¯ s ) are not necessarily of equal length. We take the vector signature g$_{a}$ (¯ s, r ) ∈ R N to be the discrete area densities of γ$_{a}$ (¯ s ) computed at each vertex. Given such a signature for fixed radius r and fixed partition ¯ s , we seek a ∗ satisfying 111 a ∗ ∈ arg min b ∈$_{R}$ 4 m ‖ g$_{b}$ (¯ s, r ) − g$_{a}$ (¯ s, r ) ‖ 2 2 s.t. γ$_{b}$ ∈ P N (4.2)