= Parabolic free surface of a rigidly rotating liquid
{title2=$z-z_0=\Omega^2r^2/(2g),\quad f=g/(2\Omega^2)$}
A liquid in rigid <rotation> about a vertical axis is at rest in its <rotating reference frame>. Constant surface <pressure> makes $gz-\Omega^2r^2/2$ constant along the <free surface>, giving the displayed paraboloid. Comparing with $z-z_0=r^2/(4f)$ gives the focal length $f$. The result assumes uniform <gravitational acceleration>, a vertical rotation axis and negligible surface-tension corrections. It explains how a rotating liquid can form a reflecting surface or a mould for a mirror.
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