Why Do Conic Sections Fail to Produce Symmetrical Ellipses?

[vector22]

I need some feedback about something that does not make sense.

The parabola and hyberbola can be found in the conic sections. These curves are seen if one looks at right angles to the plane of the section (cut surface if you will).. All math books also say that the ellipse is a result of a conic section, You know i'll be damned if I could ever find a nice ellipise from a conic section with even the least bit of symmetry to it. If you section a cone you will most often get what resembles a plane surface that has the profile of a chicken egg.

Anyway open to you comments

 

[HallsofIvy]

What you are saying is simply not true. Cut a cone at an angle to the axis and the cross section will be symmetric.

 

[dimitri151]

The section you speak of is actually more like a dinosaur egg than a chicken egg.

 

[vector22]

this morning, just to be sure, i looked through a calculus book by James Stewart and he claims that you get nice looking symmetric ellipses from conic sections. I'm sorry but I am seeing something what resembles chicken eggs or like dimitri151 says "dinosaur egg" shaped ellipses. To make things even more confusing I have done sections on cylinders (yes I've cut through wooden dowling with a chop saw) and the resulting section looks very symetric like the ellipses we see in textbooks (not sure if a cylinder section is an elipse though??)

comments?

 

[micromass]

Yes, cylindrical cuts are ellipses too. But I don't quite see why you think that conic sections are like chicken eggs? They are not... The easiest way to see this is to make a cone in real life. Cut it and see what you get. You'll get a nice symmetric ellipse!

In fact, if you cut it parallel to the base, then you get a circle!

Another way to see this is by calculating, but it's a little tougher. The equation of a cone is x²+y²=z². If you now take an arbitrary plane \alpha x+\beta y+\gamma z=\delta, and if you cut it with the cone, then you'll get a general equation for a conic section. If you graph it, then you'll see symmetries!

 

[vector22]

Micromass said:

Yes, cylindrical cuts are ellipses too.

If that is so then how can a section of a cylinder be the same shape as the section from a cone? The cylinder has only a single radius dimension while the radius of the cone varies so how can you introduce a paradox by saying that an ellipse from a cylinder section will be the exact same shape as the ellipse from a conic section?.

The only shape of section that both cones and cylinders share exactly is the circle

Expecting severe criticism at this point

 

[dimitri151]

I was being a little facetious but dinosaur eggs are symmetrical unlike bird eggs.

 

[vector22]

dimitri151

I've seen dino eggs and yes they look different. symmetry? maybe


Micromass has just said that cylinder sections are ellipses, do you agree or disagree with that?

 

[micromass]

I was being a little facetious but dinosaur eggs are symmetrical unlike bird eggs.

I don't want to hijack the thread, but aren't birds actually dinosaurs? I vaguely remember that from somewhere

 

[vector22]

In my mind any ellipse that looks like a chicken egg is not an ellipse because there is symmetry in only one dimension. The ellipses that we see in the textbooks have symmetry in 2 dimensions.

And no don't hijack this thread in order to change the subject

 

[micromass]

Well, all ellipses have two symmetry axis. So a chicken egg is indeed no ellipse.

However, the curve arising from conic sections are ellipses, not chicken eggs. It can be proved by 3D-geometry. Or you can easily see this for yourself if you actually cut a cone.

In my old university stood a scale model of cone which was cut. The curve that it delivered was indeed an ellipse with two symmetry axis.

I understand that you don't believe this, because it doesn't correspond to your intuition. However, it is certainly true. And you should try to find a way to convince yourself of this fact. Maybe you could make an experiment and actually make a small cone which you cut?

 

[vector22]

heh small cone?

take a cone with the large open end a mile in diameter and pretend you are an ant and crawl all the way down there at the bottom of the cone and you'll get no further because your ant body will meet more than one sde of the cone walls. You will notice a failry curved surface at that point. But near the open end of the cone the ant notices what seems to be a flat surface - very little curvature. A conic section from the opening of the cone all the way to the bottom of the cone where the ant is can't possibly result in a nice symetric ellipse. get a grip on reality people

 

[dimitri151]

A cylinder is just a cone with its vertex at infinity. So the section of a cylinder can be the same as a section of a cone. Whaddayasay to that?

 

[micromass]

heh small cone?

take a cone with the large open end a mile in diameter and pretend you are an ant and crawl all the way down there at the bottom of the cone and you'll get no further because your ant body will meet more than one sde of the cone walls. You will notice a failry curved surface at that point. But near the open end of the cone the ant notices what seems to be a flat surface - very little curvature.

Yes, I think I see what your problem is. The thing is that there appears to be very little curvature when you look horizontally! But if you cut the cone with a plane, then the plane cuts the cone at an angle. And this angle also contributes a small curvature. I.e. instead of looking horizontally, you'll need to look at an angle.

When you're down in the cone, then you also need to look at an angle. This will reduce it's curvature a little. I hope this was understandable, it's hard to explain.

A conic section from the opening of the cone all the way to the bottom of the cone where the ant is can't possibly result in a nice symetric ellipse. get a grip on reality people

The beauty of it is that it will result in a nice symmetric ellipse! What would it take to convince you of this fact??

 

[vector22]

will this effect my chances of getting into Cambridge?

 

[94JZA80]

i've solved your discrepancy. you see, the conical cross section that produces an ellipse also produces an egg shape as well, literally depending on your point of view. consider the following image of an elliptical conic section:

[PLAIN]http://img217.imageshack.us/img217/3148/ellipse.jpg

if you were to hover just above the vertex of the cone and look down (such that your line of sight lie directly along the cone's axis of symmetry), the cross section would appear as an egg shape, not an ellipse. but look at the image again and you'll note that i took the liberty of drawing in an axis that is perpendicular to the cross section itself. note that its not perpendicular to the horizontal, but that it still makes a greater angle with the cone's base than the side of the cone does with its base. if you take this axis to be your line of sight (instead of using the cone's axis of symmetry), you'll note that the cross section makes an ellipse that is perfectly symmetrical about both its minor and major axes.*EDIT* - sorry for the fairly non-descriptive response, but the above was the best i could do in the 5 free minutes i had at work. i wanted to elaborate a bit, so now that i have the time, let me say that conic sections were meant to be viewed with a line of sight that is perpendicular/normal/orthogonal to the plane that the conic lies in. when you view the above conic section from directly overhead (i.e. from a line of sight that is perpendicular to the plane of the conic which, incidentally, does not line up with the cone's axis of symmetry), it will always be an ellipse. it's only when your line of sight with the plane of the conic deviates from 90° that the conic appears to take on an egg shape.

 

[dimitri151]

If the ant was hovering over the vertex of the cone everything on the surface of the cone would appear as a circle. Looking straight down the axis of the cone, he would look say 30 degrees north and see one edge of the elipse, then look 30 degrees south and see the other edge. In all cases the ant would turn his head the same angle from the axis to view the elipse. I'm assuming all objects are idealized, so there is no imperfection of form to give depth reference.

 

[94JZA80]

If the ant was hovering over the vertex of the cone everything on the surface of the cone would appear as a circle. Looking straight down the axis of the cone, he would look say 30 degrees north and see one edge of the ellipse, then look 30 degrees south and see the other edge. In all cases the ant would turn his head the same angle from the axis to view the ellipse.

...not unless the ant were actually at the vertex. if he were hovering somewhere above it, he would certainly see something else. if he started at the vertex, he would observe the circle you speak of. but that's irrelevant, b/c one cannot distinguish between any of the conics while at the vertex viewing the surface edge-on - it'll always appear as a circle. but if he were to then climb the cone's axis and observe the cone below as he climbed, he would see two things. first, he would notice that the circle that is the base of the cone is slowly shrinking in diameter due to his increasing distance from it (assuming for this example that the cone is not infinite in height and actually has a base). secondly, he'd see the conic section shrinking somewhat faster than the circle that is the base of the cone itself. at first the conic would appear circular in nature, but quickly become elliptical in nature, and then very egg-like in nature as part of the conic section's edge shrinks up faster than the other parts of its edge.

i made some additional notes on the illustration i posted earlier to help visualize this:

[PLAIN]http://img233.imageshack.us/img233/3633/ellipse3.jpg


the simple fact that C is orthogonally farther from the cone's axis than D is illustrates the facts that not only is the ellipse is not centered on the cone's axis, but it doesn't even lie in a plane orthogonal to it. granted, the latter was obvious from the onset, as that's what defines an ellipse in the first place. however, as a consequence of those facts, anyone directly above the cone's vertex (but not at the vertex) will see an ellipse that only appears to be symmetrical along its major axis, or egg-like in shape. but remember, this is just from the perspective of someone above the vertex of the cone staring down. one would have to move from that position directly above the vertex of the cone to a point somewhere along the axis that orthogonally pierces the center of the ellipse (the axis that i drew in myself) in order for the ellipse to be viewed the way it was intended to be viewed - symmetric along both its major and minor axes.

at any rate, given that it can be shown that ellipses can appear egg-like in shape if not viewed on a skew or an angle, i think that maybe the OP was simply limiting his view of the elliptical conic section to the cone's axis of symmetry, and was not actually trying to look directly at it (face-on).

 

[dimitri151]

Yeah, that's was the point: the ant is at the vertex. And how do you know the ant is a he. If the ant is hovering it has wings, then it's probably a queen ant.

Somehow we lost the point. The important thing is what the section is, not what it appears to be.

 

[94JZA80]

Yeah, that's was the point: the ant is at the vertex. And how do you know the ant is a he. If the ant is hovering it has wings, then it's probably a queen ant.

my mistake...i must concede this much