In this section, we consider the case where nonasymptotic densities and first deriva- tives are known along a T-shaped set (i.e., for all s with a fixed radius ˆ r and for all r ≤ ˆ r with a fixed ˆ s ). We show that this information is sufficient to guarantee recon- structability modulo reparametrizations, translations, and rotations. 

Lemma 4.5.1. Assume that γ is TGL for ˆ r (and thus all r ≤ ˆ r ). Then if we know g ( s, r ) , g$_{s}$ ( s, r ) = ∂ g ( s,r ) ∂s , and g$_{r}$ ( s, r ) = ∂ g ( s,r ) ∂r for ( s, r ) ∈ ([0 , L ] × { ˆ r } ) ∪ ( { ˆ s } × (0 , ˆ r ]) , we can reconstruct γ ( s ) ∈ R 2 for all s ∈ [0 , L ] modulo reparametrizations, translation, and rotations. (See figure 4.10.) 

Proof: As was shown in section 4.4, g$_{r}$ gives us the length of the arc ∂D ( s, ˆ r ) ∩ Ω and g$_{s}$ tells us precisely what position this arc is along ∂D ( s, ˆ r ) with respect to the direction γ $^{′}$( s ). The assumption of TGL for r = ˆ r implies TGL for 0 < r < ˆ r (see remark 4.2.12) and this implies that γ has the 2 arc property and transverse intersections with ∂ D( s, r ) for all disks corresponding to ( s, r ) ∈ ([0 , L ] × { ˆ r } ) ∪ ( { ˆ s } × [0 , ˆ r ]). Since we care only about reconstructing a curve γ isometric to the original curve, we choose γ (ˆ s ) = (0 , 0) ∈ R 2 and γ $^{′}$(ˆ s ) = (1 , 0). Taken together, g$_{s}$ (ˆ s, r ) and g$_{r}$ (ˆ s, r ) locate both points in ∂D (ˆ s, r ) ∩ γ for all r ∈ [0 , ˆ r ]. This yields γ ∩ D( γ (ˆ s ) , ˆ r ). Now, simply increase s , sliding the center of a disk of radius ˆ r along γ ∩ D( γ (ˆ s ) , ˆ r ), using g$_{r}$ ( s, ˆ r ) to find the element of γ ∩ D( γ ( s ) , ˆ r ) outside D ( γ (ˆ s ) , ˆ r ), using the fact that the other element of γ ∩ D( γ ( s ) , ˆ r ) is inside D( γ (ˆ s ) , ˆ r ) and known. This process can be continued until the entire curve is traced out in R $^{2}$. 

