the polygon. With the side lengths identified earlier, this completely determines the polygon Ω up to translation and rotation. We will assume that γ is TGL for the radius ˆ r . We will also assume that we know the first, second, and third derivatives of g ( s, r ) for r = ˆ r . Under these assumptions, γ is generically reconstructible. By generic we mean the admittedly weak condition of density – reconstructible curves are C 1 dense in the space of C 2 simple closed curves. Theorem 4.7.1. Define G ≡ { γ | γ is a C 2 simple closed curve and TGL for r = ˆ r } . Suppose that, for r = ˆ r , for all s ∈ [0 , L ] , and for each γ ∈ $_{G}$we know the first-, second-, and third-order partial derivatives of g$_{γ}$ ( s, r ) . Then the set of reconstructible γ ∈ G is C 1 dense in G where reconstructability is modulo reparametrization, translation, and rotation. Proof: In section 4.4 we showed that ∂g ( s,r ) ∂r = r ( θ$_{2}$ − θ$_{1}$ ) and ∂g ( s,r ) ∂s = r (sin( θ$_{2}$ ) − sin( θ$_{1}$ )), where the notation is as in Figure 4.12. Because γ is TGL, we can solve for θ$_{1}$ and θ$_{2}$ from these two derivatives as in the proof of Theorem 4.6.1. Claim 1. The following equations hold: ∂ $^{2}$g ( s,r ) ∂r 2 = θ$_{2}$ − θ$_{1}$ + r ( ∂θ$_{2}$ ∂r − ∂θ$_{1}$ ∂r ) and ∂ $^{2}$g ( s,r ) ∂r∂s = sin( θ$_{2}$ ) − sin( θ$_{1}$ ) + r (cos( θ$_{2}$ ) ∂θ$_{2}$ ∂r − cos( θ$_{1}$ ) ∂θ$_{1}$ ∂r ) . Proof of Claim 1: Simply differentiate the expressions we already have for ∂g ( s,r ) ∂r and ∂g ( s,r ) ∂s . □ We wish to express this in terms of ν$_{1}$ and ν$_{2}$ . Note that if we expand the circle radius by Δ r , the right exit point s$_{+}$ ( s ) moves approximately (i.e., considering first-order terms 106 4.7 Simple closed curves are generically reconstructible using fixed radius data