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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?

Martin Sleziak
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egglog
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    http://www.matematicasvisuales.com/english/html/geometry/elipses/dandelincylinder.html – Intelligenti pauca Aug 06 '21 at 07:55
  • @Intelligentipauca Thanks, this was very insightful – egglog Aug 06 '21 at 07:59
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    This is a property of conic sections (with a cylinder being a degenerate cone). So the answer to your final question about a truncated conical cup is "yes it is an ellipse" so long as the curve does not hit the end of the cup – Henry Aug 06 '21 at 08:13
  • Related questions (I cannot yet comment): https://math.stackexchange.com/questions/2618675/prove-that-the-intersection-of-a-cylinder-and-a-plane-forms-an-ellipse#2618730 https://math.stackexchange.com/questions/3704089/showing-that-the-intersection-of-a-cylinder-and-a-plane-is-an-ellipse – William A. Noble Aug 06 '21 at 08:15

2 Answers2

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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.

v0 big base coffee cup

a0 circle

a45 ellipse

a73 (portion of) parabola

a90 (portion of) hyperbola

GEdgar
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  • The opening of the cup is the smaller circle. It is right-side up. I did this to get non-ellipse cases without spilling the coffee. – GEdgar Aug 06 '21 at 16:53
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    that is confusing, because it doesnt look like a cup at all, but it does look like an upside down cup – Zombo Aug 06 '21 at 16:58
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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 $26.5^{\circ}$).

enter image description here

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!).

enter image description here

Robert Z
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