Does tilted coffee form an ellipse?

Question

Consider a cup of coffee in a circular mug. If this mug is resting flat on a surface, the surface of the coffee forms a circle. If you tilt the cup in any direction, however, does the surface formed resemble an ellipse? If so, how do you prove it?

I drew a simple 2D representation of the problem, and although it looks as though the surface formed is in fact an ellipse, I have no simple way of proving this.

What happens if the cup you have is not simply cylindrical, but is two parallel circles of differing radius, with their centres both going through the same line perpendicular to themselves? Surely this does not form an ellipse, but then, what?

Answer 1

I agree with @Henry ... two parallel circles of different size both perpendicular to the line through their centers ... this is exactly a "conic section". The plane section is an ellipse, parabola, or hyperbola. Parabola and hyperbola would be possible without spilling if the open end of the cup is the smaller circle.

big base coffee cup

circle

ellipse

(portion of) parabola

(portion of) hyperbola

Answer 2

I have no general answer, but I tried to find a meaningful example.

Consider the surface-cup $z=(x^2+y^2)^{1/4}$ (which has a strictly concave graph) with a certain amount of coffee inside. If we tilt the cup then the surface of the coffee does not form an ellipse.

Below there is a 3D graph where the surface of the coffee is given by the intersection of the cup with the plane $2z=x+1$ (which means that we tilted the cup of about $26.5^{\circ}$).

As you can see the elongated coffee-surface is well rounded at the left end and pointed at the other side. Actually its projection on the $xy$-plane is bounded by the curve $$2(x^2+y^2)^{1/4}=x+1$$ which has a singular point at $(1,0)$ (it's not an ellipse!).