at all those intersections. Step 3: TGLL implies that there is a constant K < ∞ such that γ ∩ D( γ ( t ) , r ) is the graph of a function whose x-axis direction is parallel to γ $^{′}$( t ) and this function is Lipschitz with Lipschitz constant K . Since γ ′ is uniformly continuous, there will be a δ$_{1}$ such that if | u − v | < δ$_{1}$ , then ∠ ( γ $^{′}$( u ) , γ $^{′}$( v )) < arctan 2 K − arctan K . Define δ$_{K,s}$ = min( δ$_{s}$, δ$_{1}$ ). Define δ$_{K,r}$ = min( δ$_{γ}$, δ$_{K,s}$ cos( π/ 2 − ϵ$_{ν}$/ 3)). Then γ ∩ D( γ ( t ) , r + δ$_{K,r}$ ) is the graph of a Lipschitz function with Lipschitz constant at most 2 K when γ $^{′}$( t ) is used as the x-axis direction. That is, for all t , γ is TGLL with Lipschitz constant 2 K for disks of radius r + δ$_{K,r}$ . The result follows by lemma 4.3.1. γ ≈ H $^{1}$( ∂D ∩ Ω)Δ r ∂g ∂r = H $^{1}$( ∂D ∩ Ω) r r + Δ r 4.4 Derivatives of g ( s, r ) Figure 4.8: Deriving ∂g ∂r as the arclength of the circular segment. 100