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We will use the fact that γ always crosses ∂D transversely to prove that γ is in fact TGLL on slightly bigger disks of radius r + δ as long as one takes a somewhat bigger Lipschitz constant ˆ K . It is then an immediate result of lemma 4.3.1 that we can find an approximating polygon that is TCGL with radius r . 

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Lemma 4.3.4. If γ is TGLL with radius r , then it is TGLL with radius r + δ for some δ > 0 and there is an approximating polygon P$_{γ}$ which is TCGL with radius r . 

Proof. Step 1: Show that the quantities ν$_{1}$ and ν$_{2}$ are continuous as a function of s ∈ [0 , L ] .(see Fig. 4.6) 

Define R $^{2}$( s, t ) ≡ || γ ( s ) − γ ( t ) || $^{2}$. Taking the derivative, we get 

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Because ν$_{1}$ and ν$_{2}$ are both less than π/ 2 and γ is graph-like in the disk, we have that both elements of this derivative are nowhere zero. By the implicit function theorem, we 

