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Let $\gamma$ be a convex smooth plane curve of length $l$.
I need to compute the area of the strip swept by the outer normal segments of length $r$ to $\gamma$ and the length of the outer boundary oval of the strip. I assume that a parameterization of the curve is given. But in any case, I have no idea how to decide. I tried to do this for the circle, but it didn't work either.

WIT
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  • For arcs of the same length on circles of different sizes, the area is different, so more information is needed than the values of $\ell$ and $r. \qquad$ – Michael Hardy May 25 '20 at 15:43
  • @Michael Hardy It's a bounded curve. So values of $ℓ$ and $r$ are enough. – WIT May 25 '20 at 15:51
  • If the curve is closed, the area of the strip is $\ell r + \pi r^2$, see answers of this and that. – achille hui May 25 '20 at 16:02
  • @WIT : I'm guessing you meant it is a closed curve. But your question doesn't say that. – Michael Hardy May 25 '20 at 16:08
  • @Michael Hardy: yes. I got it wrong. I meant closed curve. – WIT May 25 '20 at 16:13
  • @achille hui: thank you! – WIT May 25 '20 at 16:13
  • If a parameterization of $\gamma(t)$ is given, then you can find a parameterization of the outer curve as $\eta(t)=\gamma(t)+r N(t)$, where $N(t)$ is the unit normal to $\gamma(t)$ and find the enclosed area by say Green's theorem. – Marco May 25 '20 at 18:26

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