D ( p, r ) p p D ( p, r ) p D ( p, r ) Figure 4.4: Additional points of intersection violate the TCGL condition. Suppose | ∂ Ω ∩ C ( p, r ) | > 2; that is, there are other points of intersection. Letting q denote one of these, there are two cases to consider (illustrated in Figure 4.4). As ∂ Ω is tangent-cone graph-like at q , then ∂ Ω ∩ C ( q, r ) is a graph in every orien- tation in the tangent cone of ∂ Ω at q . In particular, note that the tangent line to C ( p, r ) at q is in this cone. However, the line from p to q is normal to this line and thus ∂ Ω ∩ C ( q, r ) is not graph-like in this orientation, a contradiction. Therefore, this case cannot occur. This argument applies to all points in ∂ Ω ∩ C ( p, r ) so we immediately have the result that ∂ Ω always crosses C ( p, r ) transversely. There exists q ′ ∈ ∂ Ω ∩ C ( p, r ) such that there is a path along ∂ Ω in D ( p, r ) from q to q $^{′}$. That is, there exist s$_{1}$, s$_{2}$ ∈ [0 , L ) (without loss of generality, s$_{1}$ < s$_{2}$ ) such that γ ( s$_{1}$ ) = q , γ ( s$_{2}$ ) = q ′ and the image of [ s$_{1}$, s$_{2}$ ] under γ is contained in D ( p, r ) (but does not include p , since it is on another arc and ∂ Ω is simple). Thus γ enters C ( p, r ) at s$_{1}$ and exits at s$_{2}$ . If we can find s ∈ [ s$_{1}$, s$_{2}$ ] and w in the tangent cone of ∂ Ω at γ ( s ) satisfying 〈 w, p − γ ( s ) 〉 = 0, we will contradict that ∂ Ω is tangent-cone graph-like. 87 (a) ∂ Ω does not cross C ( p, r ) at q . (b) ∂ Ω crosses C ( p, r ) at q .