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?
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.
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.