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This oval is made up of 4 arcs, 2 on the left and right sides of radius 1 and 2 on top and bottom of radius $R$. Given that the the oval fits in a $4 \times 8$ rectangle, is it possible to find $R$ ?

oval in rectangle

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    @Heike: The curve isn't smooth in the technical sense, only $C^1$.2011-12-06

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Let $L$ be the length of the box, and $H$ be its height. By Pythagoras' theorem we have $(R-1)^2=((L-2)/2)^2+(R-H/2)^2$ from which it follows that $R=\frac{(L-2)^2+H^2-4}{4(H-2)}$ For this particular example we have $L=8$, $H=4$, and $R=6$.

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    @schooler: The solution assumes continuous tangents. That implies that the centres of the circles must lie in the same direction from the transition points. So the centres of the big circles lie at the intersections of the lines through the centres of the small circles and the transition points. The triangle that Heike applied Pythagoras' theorem to is one formed by a centre of a small circle, a centre of a big circle and the centre of the rectangle.2011-12-07