content_image/313394.jpg
ν$_{2}$ ( s ) similarly. We have ν$_{1}$, ν$_{2}$ ∈ ( − π $_{2}$, π $_{2}$) due to the fact that all circle crossings are transverse by theorem 4.2.1. 

When the proper s to use is implied by context, we will often simply write s $^{+}$, s $^{−}$, θ$_{1}$ , θ$_{2}$ , ν$_{1}$ and ν$_{2}$ in place of s $^{+}$( s ), s $^{−}$( s ), and so forth. 

## 4.2.4 Calculus on Tangent Cones 

The following result is a version of the intermediate value theorem for elements of the tangent cones. 

Lemma 4.2.13. Suppose ∂ Ω is tangent-cone graph-like on D ( γ ( s ) , r ) and s$_{1}$ < s$_{2}$ such that γ ( s$_{1}$ ) , γ ( s$_{2}$ ) ∈ D ( γ ( s ) , r ) . Further suppose that w$_{1}$ ∈ T$_{γ}$ ( s$_{1}$ ) , w$_{2}$ ∈ T$_{γ}$ ( s$_{2}$ ) , α ∈ (0 , 1) , and let w ′ = αw$_{1}$ + (1 − α ) w$_{2}$ . Then, there exists s ′ ∈ [ s$_{1}$, s$_{2}$ ] such that either w ′ or − w ′ is in T$_{γ}$ ( s $^{′}$) . 

Proof. Let n be a unit vector in R 2 with n ⊥ ( αw$_{1}$ +(1 − α ) w$_{2}$ ). We have α 〈 n, w$_{1}$ 〉 = − (1 − α ) 〈 n, w$_{2}$ 〉 . It suffices to consider only 〈 n, w$_{1}$ 〉 ≤ 0 ≤ 〈 n, w$_{2}$ 〉 as the argument is identical in the other case. Note that since 0 ≤ 〈 n, w$_{2}$ 〉 = c$_{1}$ 〈 n, lim$_{t}$$_{↑}$$_{s}$$_{2}$ γ $^{′}$( t ) 〉 + c$_{2}$ 〈 n, lim$_{t}$$_{↓}$$_{s}$$_{2}$ γ $^{′}$( t ) 〉 for some nonnegative constants c$_{1}$, c$_{2}$ not both zero, at least one of the inner products on the right is nonnegative. Using the notation of definition 4.2.4, we define M$_{2}$ = argmax$_{Γ}$$_{∈{}$$_{Γ}$ $^{+}$, Γ $^{−}$} 〈 n, Γ 〉 and have 〈 n, M$_{2}$ 〉 ≥ 0. We similarly define M$_{1}$ with respect to w$_{1}$ such that 〈 n, M$_{1}$ 〉 ≤ 0. 

Define 

<img src='content_image/313394.jpg'>

and φ ( t ) = 〈 n, v ( t ) 〉 . Since φ ( s$_{1}$ ) ≤ 0 ≤ φ ( s$_{2}$ ), the argument proceeds as in theorem 4.2.1 to yield ¯ s ∈ [ s$_{1}$, s$_{2}$ ] and ¯ w ∈ T$_{γ}$ (¯ s ) such that 〈 n, ¯ w 〉 = 0. Thus ¯ w = kw ′ for some 

