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Finally, if there is not a distinct maximum, we can choose one of the maxima and add a small twist to the curve at that point. See Figure 4.14. The idea is that a small twist, applied to the leading edge of the tangents we are comparing to get the turning angle, will increase the angle most at the center of the twist. If this corresponds to a nonunique global maximum, we end up with a unique global maximum. 

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Now the correspondence scheme works. That is, we know that the global maximums must match, and because the turning angle curves are monotonic between isolated crit- ical points, we can find the homeomorphisms in s that move the turning angle curves into correspondence. □ 

