r away from the center of the disk. Now we use this correspondence between the θ + ν curves to solve for s $^{−}$( s ) and s $^{+}$( s ). But these curves can differ by a homeomorphism of the domain. Thus, we can only find the correspondence if there is a distinguished point on those curves as well as no places where the values attained are constant. The turning angle curves having isolated critical points and a unique maximum or minimum is sufficient for our purposes. To get isolated extrema, start by approximating the curve γ with another one, ˆ γ , that agrees in C 1 at a large but finite number of points { s$_{i}$ } N i =1 (i.e. agrees in tangent direction as well as position) and has isolated critical points in the derivative of the tangent direction. Now perturb ˆ γ to one that is C 1 close (but not C 2 close) by using oscillations about the curve so that the 2nd and 3rd derivatives are never simultaneously below the bounds on the 2nd and 3rd derivatives of the curve we started with. We do this in a way that alternates around the curve. See Figure 4.13. In a bit more detail, suppose that max { d $^{2}$ˆ γ/ds $^{2}$, d$^{3}$ˆ γ/ds $^{3}$} < L$_{1}$ . Choose a starting point on the curve; s = 0 works. Now begin perturbing ˆ γ at the point s$_{ˆ}$ r in the positive s direction such that | ˆ γ ( s$_{ˆ}$ $_{r}$) − ˆ γ (0) | = ˆ r . We name the newly perturbed curve ˆ γ and we keep L$_{1}$ < max { d 2 ˆ γ/ds $^{2}$, d3 ˆ γ/ds $^{3}$} < L$_{2}$ . We continue perturbing until we have reached s$_{2}$$_{r}$ defined by | ˆ γ ( s$_{2ˆ}$ $_{r}$) − ˆ γ ( s$_{ˆ}$ $_{r}$) | = ˆ r . We begin perturbing again when we reach s$_{3ˆ}$ $_{r}$. Continue in this fashion around ˆ γ . The last piece, shown in green in the figure, will require a perturbation that is distinct in size due to the fact that it will interact with the perturbation that starts at s$_{ˆ}$ $_{r}$. On this last piece, we enforce L$_{2}$ < max { d 2 ˆ γ/ds $^{2}$, d3 ˆ γ/ds $^{3}$} < L$_{3}$ . All these perturbations can be chosen with isolated singularities in derivatives, thus giving us θ + ν curves that are monotonic between isolated singularities. (In fact, we might as well choose all perturbations to be piecewise polynomial perturbations. This immediately gives us the isolated singularities and monotonicity that we want.) 109