Interesting question: pouring water out of a glass

[jemidiah]

Find a depth function D(x,y) which gives the distance through the cylinder you would travel if you shot through it at the indicated point, looking side-on as in your picture. Integrate over the function in 2 dimensions on the area where water should be for a given theta piecewise--the limits of integration wouldn't be bad in Cartesian. This integral gives the volume of water as a function of theta. Since volume and height are equivalent information, invert to find theta as a function of height.

The trick is to find the depth function D for a given theta. I suppose it's not too difficult after all. Find one for theta = pi/2 (straight up and down), and rotate that function in the plane through a rotational transformation about the lower right pivot. The straight up and down version would be straightforward--it would just be a function giving the length of parallel secants cutting through a circle, which can be derived without much trouble geometrically.


None of the individual pieces I've listed seems overly complicated, but putting it all together would be a headache, and would result in some awful piecewise solution with trig functions flying around everywhere.

I thought of two other solution methods. The first is to brute force the volume through an integral in cylindrical coordinates, but that requires very complicated boundary equations. The second is to use dimensional analysis, but the angle is unitless and could show up in an arbitrarily complicated way. The above seems about as simple as you'll get, though of course there may be some unforeseen and magical shortcut .