Exploring Conic Sections in Everyday Objects: The Case of a Cup of Tea

[3.141592]

Homework Statement

I was sitting drinking a cup of tea earlier. The cup was of course cylindrical. I was just gazing into the cup looking at the tea as my cup was flat and the top level of the tea looked like a circle. Watching the tea as I tilted the cup to drink I noticed that the shape of the top level of tea changed in the way a circle might if you stretched it. Is it the case that the shape of the tea moves from a circle, when my cup is upright, through a parabola and then an ellipse?

I am sorry I do not know anything about conic sections other than that you can get them by slicing through cones at various angles. Having looked them up online, I can't tell without some equations representing the shape made by my tea, if the curves are either of these conic sections. I don't know enough to say just by looking and I don't know if the parabola or the ellipse comes first i.e. with a slight tilt of the cup.

Obviously a very full cup cannot tilt much without spillage so let's assume any old cylindrical cup that is half full.

Thanks in advance.

 

[3.141592]

Ok I see now that the ellipse comes before the parabola in terms of the steepness of the tilt.

 

[rude man]

Here be the answer to your query. (It's an ellipse. The cutting plane stays horizontal while tilting the cup).

http://en.wikipedia.org/wiki/Cylinder_(geometry )

 

[gneill]

I'm pretty sure the only curves you can get are circles, ellipses, and straight lines. That's assuming you replace the cup with a cylinder of infinite length so you don't "run out of cup" at large tilt angles

Parabolas have "arms" that become parallel in the limit. There's no cross section of a circular cylinder where that can happen, except for cuts parallel to the long axis which produces two separate (but parallel) lines.

 

[3.141592]

Thanks!

I didn't understand (from Wiki): "A plane tangent to the cylinder, meets the cylinder in a single straight line. Moved while parallel to itself, the plane either does not intersect the cylinder or intersects it in two parallel lines. All other planes intersect the cylinder in an ellipse or, when they are perpendicular to the axis of the cylinder, in a circle."

A plane being parallel to itself? Does this mean the sort of thing that happens if you slide around an upright dvd case on your bookshelf, it moves position but remains parallel to its previous position?

Anyway, that wasn't my homework question so if no one answers it's ok. Now I know my tea moves from a circle to the motion of planetary orbit! :)

 

[3.141592]

Thanks gneill. These arms that become parallel in the limit: does this mean, if I were to look at the parabola and zoom out to some hypothetical far away distance, or draw it and make the y-axis up to infinity or some such? Sorry, as you can see, I don't know much about maths.

Does the situation change if my cup isn't a cylinder but a truncated cone? So the top is a circle, but then is gets narrower down to the base. But the base is not a sharp point of course, more like the cup was a cone and got sliced horizontally nearer the narrow end. Still all ellipses? It's a normal length cup.

 

[rude man]

Thanks!

I didn't understand (from Wiki): "A plane tangent to the cylinder, meets the cylinder in a single straight line. Moved while parallel to itself, the plane either does not intersect the cylinder or intersects it in two parallel lines. All other planes intersect the cylinder in an ellipse or, when they are perpendicular to the axis of the cylinder, in a circle."

A plane being parallel to itself? Does this mean the sort of thing that happens if you slide around an upright dvd case on your bookshelf, it moves position but remains parallel to its previous position?

Anyway, that wasn't my homework question so if no one answers it's ok. Now I know my tea moves from a circle to the motion of planetary orbit! :)

The plane is kept in its original orientation but moved into the cylinder parallel to the cylinder's axis. I agree the wording here is very bad.

Anyway, that's not your part. Your part is "All other planes ... in an ellipse ..."

 

[gneill]

Thanks gneill. These arms that become parallel in the limit: does this mean, if I were to look at the parabola and zoom out to some hypothetical far away distance, or draw it and make the y-axis up to infinity or some such? Sorry, as you can see, I don't know much about maths.

Yes, that's the idea.

Does the situation change if my cup isn't a cylinder but a truncated cone? So the top is a circle, but then is gets narrower down to the base. But the base is not a sharp point of course, more like the cup was a cone and got sliced horizontally nearer the narrow end. Still all ellipses? It's a normal length cup.

Well, the geometry of sections through such a surface is going to depend upon what profiles it passes through. When you stray from simple symmetries like cones or cylinders, you can expect things to get trickier. If you cut such a shape with a horizontal plane you will still get circles, and if you stay away from the truncated base you'll get ellipses. Once the truncated portion comes into play, the shape will depend upon the details of its curvature.

With the right cuts through the cup you could get parabolas and half hyperbolas too.

 

[3.141592]

The plane is kept in its original orientation but moved into the cylinder parallel to the cylinder's axis. I agree the wording here is very bad.

Anyway, that's not your part. Your part is "All other planes ... in an ellipse ..."

Ah ok, thanks, I understand now!

 

[3.141592]

Yes, that's the idea.

Well, the geometry of sections through such a surface is going to depend upon what profiles it passes through. When you stray from simple symmetries like cones or cylinders, you can expect things to get trickier. If you cut such a shape with a horizontal plane you will still get circles, and if you stay away from the truncated base you'll get ellipses. Once the truncated portion comes into play, the shape will depend upon the details of its curvature.

With the right cuts through the cup you could get parabolas and half hyperbolas too.

That's great thanks. So every time I take a sip of tea I can have a peek at the shape drawn out by a planet. Neat.

 

[haruspex]

You get a parabola from a cone if, and only if, the plane is parallel to one flank of the cone (and not through the vertex).
But your original question is in the context of perspective, which is in principle a bit different. It is far from obvious that a circle viewed obliquely also produces an ellipse.
You may also have noticed interesting patterns of light in the base of an empty cylindrical cup when lit obliquely. A cardioid is formed.