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Imagine a spherical ball with radius r that equally coated into two hemisphere P(grey) and S(white) is thrown at the pool. Consider the ball can float (up-down) and rotate. There is an interface separate two different fluid A(air) and B(water). Assume that the interface is flat. α defines the position of the ball and β defines the orientation of the ball. So α larger means ball is more in zone A and vice versa. Range of β and α is $[0,360^0]$ and $[0,180^0]$. The Schematic representation of this is below. enter image description here

To calculate the surface area of P(grey) coated portion of the ball into A(air) is given by following equation, $\mathrm{Area}_{P@A}= r^2\int_{\theta=\frac{\pi}{2}-\beta}^\alpha \int_{\phi=\arcsin(1/(\tan\theta \tan \beta))} ^{\pi -\arcsin(1/(\tan\theta \tan \beta)} \sin\theta\; d\theta d\phi.$

After solving this equation, we can get the following solution. $\textrm{Area}_{P@A}= 2 r^2 \left\{ \cos (\alpha) \sin ^{-1}(\cot (\alpha) \cot (\beta)) -\tan ^{-1}\left(\frac{\cos (\beta)}{\sqrt{\sin ^2(\beta)-\cos ^2(\alpha)}}\right)\right\}+\pi r^2 (1-\cos (\alpha)).$

Note that the solution is defined as a function of α and β. If we change the value of α and β, the area of P coated portion into A(air) will change eventually. Thus we can calculate the area for all valid condition of $\alpha$ and $\beta$.

Now consider a different situation where P and S are not equally coated rather S>P. To show that I have introduced another angle γ also $r1$ the radius of the base of the cap. enter image description here

Now how can i write $z$ as an increased angle? what will be the limits of integration for a spherical cap? Is there any particular analytical solution exists to calculate the area P(grey) into A(air) as a function of $α,γ,β$?

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marked as duplicate by Ted Shifrin, Xander Henderson, воитель, The Count, YuiTo Cheng Jul 9 at 1:01

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