I am trying to derive the following equation from a paper I am studying, which the author has derived from the graph above. The two slightly curved lines here are modelled as the surfaces of spherical caps, with surface area S (known).
The equation is
$$\frac{X}{4\sqrt{S/\pi}} = \sin( \phi_0) + \frac{\sqrt{S/ \pi}}{R}\cos( \phi_0).$$
I am thinking that if the spherical cap is flattened out, the curved line can be seen as the diameter (D) of this flattened circle, hence $D=2\sqrt{S/\pi}$. I believe this is where the $\sqrt{S/\pi}$ term comes from.
As for the rest, I have managed to get an equation for $D$ in terms of $\phi$, $\alpha$ and $D$, using that $D=R\theta$ (as the triangle in the diagram can be seen as a segment of a circle).
My solutions however all seem pretty complicated and I cannot manage to get them to match up to the correct one. Where am I going wrong? Do I need to use a different approach? (I thought about using the volume of revolution but I can't work out how to use that here).
Can anyone help?
Best Answer
This doesn't seem quite right to me. If I take $S$ to be half the surface area (i.e., the surface area of one of the "arcs"), then I have $S=4\pi R^2\sin^2\alpha$, so $\sqrt{\frac S\pi} = 2R\sin\alpha$. By similar triangles, the angle between the arc of the circle and the chord joining the endpoints of that arc is $\alpha$ as well, and so $$\frac x2 = 2R\sin\alpha\sin(\phi+\alpha) = 2R\sin\alpha\big(\sin\phi\cos\alpha + \cos\phi\sin\alpha\big).$$ Since $\alpha$ is presumably small, we take $\cos\alpha\approx 1$, and this becomes $$\frac x{2\sqrt{S/\pi}} \approx \sin\phi + \cos\phi\sin\alpha = \sin\phi+\frac{\sqrt{S/\pi}}{2R}\cos\phi.$$ This is "close" to what you posted, but certainly different. At this point, I'm not sure the author is correct.