a polygonal approximation to ∂ Ω that is TCGL with radius r − ϵ and such that every point on ∂ Ω is within distance ϵ 6 of the polygon. Proof. First, choose a finite number of points along the boundary such that the arc length along γ between any two neighboring points is no more than ϵ $_{3}$. These will be the vertices of our polygon. Similarly to γ , we let φ be an arclength parameterization of this polygon so that they both encounter their common points in the same order. The fine spacing between vertices guarantees that we obtain the ϵ 6 bound. Indeed, given any point p ∈ ∂ Ω and its neighboring vertices v$_{1}$ and v$_{2}$ , the arc length along ∂ Ω from v$_{1}$ to p plus that from p to v$_{2}$ is at most ϵ 3 by assumption. Since Euclidean distance is bounded above by arc length, we have d ( p, v$_{1}$ ) + d ( p, v$_{2}$ ) ≤ ϵ $_{3}$. This bound in turn implies that at least one of d ( p, v$_{1}$ ) and d ( p, v$_{2}$ ) is bounded above by ϵ $_{6}$. Consider a point p = φ ( t ) on a side of the polygon (i.e., not a vertex) and its neighboring vertices v$_{1}$ = φ ( t$_{1}$ ) = γ ( s$_{1}$ ) and v$_{2}$ = φ ( t$_{2}$ ) = γ ( s$_{2}$ ) (chosen with t$_{1}$ < t < t$_{2}$ and s$_{1}$ < s$_{2}$ ). By lemma 4.2.14, there exists s ∈ ( s$_{1}$, s$_{2}$ ) such that v$_{2}$ − v$_{1}$ ∈ T$_{γ}$ ( s ). Note that this is the only member of T$_{φ}$ ( t ) up to positive scalar multiplication. Combining the arcs along γ and φ between v$_{1}$ and v$_{2}$ , we obtain a closed curve with total length at most 2 ϵ 3 , so that the distance between any two points on the curve is at most ϵ $_{3}$. That is, for any s ′ ∈ [ s$_{1}$, s$_{2}$ ] and t ′ ∈ [ t$_{1}$, t$_{2}$ ], we have d ( γ ( s $^{′}$) , φ ( t $^{′}$)) ≤ ϵ $_{3}$. Let x ∈ D ( φ ( t ) , r − ϵ ). Then d ( x, γ ( s )) ≤ d ( x, φ ( t )) + d ( φ ( t ) , γ ( s )) ≤ r − 2 ϵ 3 so that D ( φ ( t ) , r − ϵ ) is contained in D ( γ ( s ) , r − 2 ϵ 3 ). Let a, b be distinct points on the polygon in D ( φ ( t ) , r − ϵ ) and consider the line connecting them. This line also intersects a $^{′}$, b′ on γ such that we have a ′ = b $^{′}$, d ( a, a $^{′}$) ≤ ϵ 3 and d ( b, b $^{′}$) ≤ ϵ 3 so that a $^{′}$, b′ ∈ ∂ Ω ∩ D ( γ ( s ) , r ). As a − b = c ( a ′ − b $^{′}$) for some scalar c > 0, we have 96 〈 v$_{2}$ − v$_{1}$, a − b 〉 = c 〈 v$_{2}$ − v$_{1}$, a ′ − b $^{′}$〉 ̸ = 0