Water Surface, Conic sections, cylinder cross-sections I've been looking at a picture of water stirred in a glass, and the shape the water forms at the surface looks like an elliptic disk. I know that if we have a cone and we cut it sideways then it forms an elliptic disk. Is this also true for cylinders? What shape is this? Is it an ellipse with (or without) transformations?

 A: Assume that the cylinder has radius $r$ and the angle of tilt of the water is $\alpha$  to the horizontal. 
Before tilting, the water surface is a circle of radius $r$ with equation $x^2+y^2=r^2$.
Assume WLOG that tilt occurs along the $x$-axis. 
A point $(r \cos\theta,r\sin\theta)$ on the circle is then mapped to the $(\frac {r\cos\theta}{\cos\alpha},r\sin\theta)$ on the new surface. The $y$-coordinate remains unchanged, but the $x$-coordinate gets stretched. 
Hence the equation of the new surface is $$\frac {x^2}{\left(\frac {r}{\cos\alpha}\right)^2}+\frac{y^2}{r^2}=1$$
which is an ellipse with semi-major and semi-minor axes $\frac r{\cos\alpha}$ and $r$ respectively.
A: 
We have to set the two together for comparison. Cylinder intersection is an ellipse with semi-axes related as $ b = a \sin \beta $
The red cylinder and blue cone ( whose axes of rotational symmetry are vertical) are cut by a common plane to produce different ellipse intersections with same major axis  $2 a$ (marked green) but different minor axes $b_1,b_2$ and different eccentricities $e_1,e_2 $ depending on cone semi-vertical angle $\alpha $.
Let us see how conics of different eccentricity arise on intersection of a cone with a plane.
$$ \boxed{e= \frac{\cos \beta}{ \cos \alpha }} $$
$$ \beta > \alpha ,\, \cos \beta < \cos \alpha .... e_1<1   .... ellipse $$
$$ \beta = \alpha ,\, \cos \beta = \cos \alpha .... e=1   .... parabola $$
$$ \beta < \alpha ,\, \cos \beta > \cos \alpha .... e>1   .... hyperbola $$
For case of cylinder $ \alpha =0,  e_2 <1 $ with common major axis $=2a$
Cone
$$ b_1= a \sqrt{1-e_1^2} $$
Cylinder
$$ b_2 =  a \sqrt{1-e_2^2} = a \sin \beta $$
which are in general different in magnitude.
It takes a 3D image to visualize different minor axes fully.
