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# Cylindrical Earth

Circles are too flat. What if we want to make a cylindrical Earth? We can take the same sort of approach, but we need a couple more parameters. Let’s say we have a cap radius $R_c$ and height $H_c$, then we can make this:

With the sphere radius $R_s$, by matching the surface areas, the possible dimensions are restricted to $2 \pi R_c^2 + 2 \pi H_c = 4 \pi R_s^2$

This one is easiest to do integrally. For the circular cap:

$\cos(\phi) = 1-\frac{r^2}{2 R_s^2}$

This defines the corner $\phi$ in terms of the cap radius. Next, for the vertical portion, have $z$ be the distance along the side of the cylinder. We find: $2 \pi \int_{\phi_c}^{\phi} R_s^2 \sin(t) dt = 2 \pi R_c z$, yielding $\cos(\phi) = 1- \frac{R_c (z+1/2)}{R_s^2}$

And the bottom is basically like the top.

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