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Let $a(t)$ and $b(t)$ be two smooth, nested convex curves in the plane, $t\in[0,1]$:
      Nested Curves
Suppose the parametrization of $a()$ and $b()$ is such that $\dot{a}(t)$ is parallel to $\dot{b}(t)$: at time $t$, the tangents at $a(t)$ and $b(t)$ are parallel. Let $ab(t) = b(t)-a(t)$ be the vector from $a(t)$ to $b(t)$, and let $\theta(t)$ be the counterclockwise angle from $ab(t)$ to $\dot{a}(t)$. I would like to claim that

$\theta(t) = \pi/2$ at least twice within $t\in[0,1]$.

My proof of this is inelegant, relying on the length $|ab(t)|$, essentially showing that if, e.g, $\theta(t) < \pi/2$ for all $t$, then $|ab(1)| >|ab(0)|$ (contradicting $|ab(1)| = |ab(0)|$). But I feel there might be a clever way to achieve this via the mean-value theorem that I am not seeing. Also, perhaps "twice" can be replaced by "four times," and perhaps "nested" need not be assumed. So I am seeking a cleaner proof that may yield further insights. Thanks for contributing ideas or pointing me in the right direction!

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1 Answer 1

up vote 5 down vote accepted

Take points $x, y\in b$ which minimize (correspondingly maximize) the function $\mathop{\rm dist}_a$. Let $\bar x$ and $\bar y\in a$ be the closest points to $x$ and $y$ correspondingly.

Clearly $\bar x=a(t)$, $x=b(t)$ and $\bar y=a(\tau)$, $y=b(\tau)$ for some $t$ and $\tau$ and $\theta(t)=\theta(\tau)=\tfrac\pi2$.

If $a$ and $b$ are non-concentric circles then you get only these two values.

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Excellent---Thanks! –  Joseph O'Rourke Jan 21 '12 at 13:45
@Anton: However, note there are four spots in my posted example at which $\theta(t)=\pi/2$. Not sure what you mean by "then you get only these two values." –  Joseph O'Rourke Jan 21 '12 at 14:09
Take two circles, say $a(t)=(1+\cos t,\sin t)$ and $b(t)=10{\cdot}(\cos t,\sin t)$. Then $\theta(t)=\tfrac\pi2$ iff $t=0$ or $\pi \pmod {2{\cdot}\pi}$. –  Anton Petrunin Jan 21 '12 at 14:49
A confusion might arise from "not concentric circles" as Anton meant here "If $a$ and $b$ are non-concentric circles then you get only these two values". –  Tapio Rajala Jan 21 '12 at 14:58
@Tapio: Thank you for clarifying! –  Joseph O'Rourke Jan 21 '12 at 15:20

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