obtain ¯ λ ∈ (0 , 1) such that 〈 ψ ( ¯ λ ) , p − γ (¯ s ) 〉 = 0. Letting w = ψ ( ¯ λ ), we obtain our contradiction. Therefore, there are no other points of intersection and | ∂ Ω ∩ C ( p, r ) | = 2. Definition 4.2.9. We say that Ω has the two-arc property for a given radius r if for every point p ∈ ∂ Ω , we have that D ( p, r ) divides ∂ Ω into two connected arcs: ∂ Ω ∩ D ( p, r ) and ∂ Ω \ D ( p, r ) . Instead of considering how D ( p, r ) divides ∂ Ω , we can equivalently frame the definition in terms of how ∂ Ω divides C ( p, r ) . That is, Ω has the two-arc property if the circle C ( p, r ) is divided into two connected arcs by ∂ Ω for every p ∈ ∂ Ω . Corollary 4.2.10. If Ω is tangent-cone graph-like for some radius r , then it has the two-arc property. Proof. This is a trivial consequence of Theorem 4.2.1. Corollary 4.2.11. If Ω is tangentially graph-like for some radius r , then it has the two-arc property for radius r . Remark 4.2.12. While the assumption of the two-arc property for disks of radius r = ˆ r does not imply the two-arc property for all r < ˆ r (see Figure 4.5), it is the case that TGL for r = ˆ r does imply that γ is TGL for all 0 < r < ˆ r . The fact that γ is TGL for all 0 < r < ˆ r follows easily from the definition of TGL and the fact that D ( p, r ) ⊊ D ( p, ˆ r ) . Suppose that ∂ Ω is tangent-cone graph-like with radius r and we have some s ∈ [0 , L ) such that ∂ Ω is tangentially graph-like at γ ( s ) with radius r . Since ∂ Ω is TGL at γ ( s ), it has two points of intersection with C ( γ ( s ) , r ) by theorem 4.2.1. In the orientation 89 4.2.3 Notation