Conic sections-The ellipse

The ellipse is one of the four types of conic sections, along with the parabola, the circle and the hyperbola. They are called conic sections because they are formed by the section of a (flat) surface onto a cone or a system of two cones united at their pinnacle points with that point being on the axis of every symmetry in their system.

The following is a drawing of an ellipse:

http://stemsoup.files.wordpress.com/...-ellipse-2.gif

Conic sections seem to have been first examined by Apollonios of Pergamon, a large city in the north of the asia minor aegean coast, near to where Troy was supposed to have existed in the past ages. Apollonios lived during the first half of the Hellenistic era, born in the middle of the 3rd century BC.

Cones- along with sections in them- were also studied by Archimedes, as well as other objects of three dimensions, mostly those formed by a rotation of a surface (eg a cube rotating so as to form a cylinder, or different types of at least isoscelic or right-angled triangles roating to some degree so as to form a cone).

While the ellipse is usually drawn as a surface, and being symmetrical to both (cartesian) axis, in reality none of its points exists on the same surface, since it is a section on a three dimensional object. So it does not really have a symmetry of a double kind, given that only in relation to its two bow-like parts which are equal and symmetrical does it have this quality. The other two bow-like parts are not symmetrical, nor equal in length, since ever point in one of them is always on a different level than any point on the other. So the only axis which is real in an ellipse is the axis of the actual cone it was formed onto.

Recently i was quite drawn into this part of geometry. Yesterday i finished a short story (around 7 pages) centered on a narrator who is of the view he had been travelling endlessly on an elliptical course, but then stopped, and now is lost.

I would be interested in asking you if you view the elliptical shape as a symbol of anything. Also you can of course reflect on the actual historic parts of the OP, or the ellipse being used in current science.

Try looking at Kepler's use of the ellipse to describe planetary orbits, based on Tycho Brahe's detailed observations of the movement of the planets. It's intriguing how Kepler got to that understanding, a step on from the Copernican & Ptolemaic idea of circular orbits.

I surely will...Thanks :)

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Originally Posted by mal4mac

Try looking at Kepler's use of the ellipse to describe planetary orbits, based on Tycho Brahe's detailed observations of the movement of the planets. It's intriguing how Kepler got to that understanding, a step on from the Copernican & Ptolemaic idea of circular orbits.

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Brahe made Kepler his assistant in order to give him all the data. No intrigue. It ended in Newton importing Galileo, developing the calculus and Halley's accurate prediction of the behavior of the comet. There is a lot more to an ellipse than a conic section. All orbits, including those of galaxies are elliptical without exception. Otherwise tangential escape from constant falling would be impossible.

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Originally Posted by cafolini

Brahe made Kepler his assistant in order to give him all the data. No intrigue. It ended in Newton importing Galileo, developing the calculus and Halley's accurate prediction of the behavior of the comet. There is a lot more to an ellipse than a conic section. All orbits, including those of galaxies are elliptical without exception. Otherwise tangential escape from constant falling would be impossible.

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++++++++++++++++++++++++++++++++++++++++++
intriguing present participle of in·trigue (Verb)
Verb
1. Arouse the curiosity or interest of; fascinate.
2. Make secret plans to do something illicit or detrimental to someone.
++++++++++++++++++++++++++++++++++++++++++

I was using it in sense 1., so *yes* intrigue. But it was worth highlighting the ambiguity :) I wasn't really trying to imply sense 2. consciously, but maybe my subconscious was prodding me.

Quick Google search reveals:

"...he went to work with Tycho in 1600. Tycho died the next year, Kepler stole the data, and worked with it for nine years." http://galileoandeinstein.physics.vi...ures/tycho.htm

Looks like sense 2. intrigue to me.

And then there's: "Did Johannes Kepler Murder Tycho Brahe?" http://hnn.us/article/144040. Now that's even more intriguing!

Yes the whole story from Kepler's laws of planetary motion to Newton's universal law of gravitation is fascinating, with the ellipse central to the story. Another intriguing (as in fascinating!) fact is that the perihelion of the orbit of mercury was observed to precess, and the amount of precession could only be explained, sensibly, by Einstein's General Theory of Relativity. Interestingly this means the ellipse remained the central feature of an orbit, Einstein's theory didn't demote the ellipse and promote some newer orbital shape.

Only an ellipse can precess, but a circle cannot, so it's another fascinating feature of an ellipse.

An ellipse also has two foci, unlike the circle with only one focus, and the Sun is at one of those foci in relation to the planetary orbits.

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Originally Posted by cafolini

There is a lot more to an ellipse than a conic section.

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I am not entirely sure what you tried to mean by that strange sentence there. It reads a bit like saying "there is a lot more to a circle than the 360degree rotation of its radius". Well, yeah, but no at the same time, given that this is what completely forms the actual shape in question. No one really stated that somehow the ellipse has a meaning only as a section on a cone, but surely an ellipse is also always a possible section of a flat surface onto a cone.

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Originally Posted by Kyriakos

I am not entirely sure what you tried to mean by that strange sentence there. It reads a bit like saying "there is a lot more to a circle than the 360degree rotation of its radius". Well, yeah, but no at the same time, given that this is what completely forms the actual shape in question. No one really stated that somehow the ellipse has a meaning only as a section on a cone, but surely an ellipse is also always a possible section of a flat surface onto a cone.

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I'm not interested in learning from anyone who's not entirely sure of anything. And I never try to mean. I mean. Good day. LOL

Sorry man, i mean i was not entirely sure just how trollish you are. I am now though so don't worry ;)

@Mal4Mac: Is one of the two focal points of an Ellipse always in the same position as the axis of the cone the ellipse is a section of? For it appears to be so:

http://upload.wikimedia.org/wikipedi...uction.svg.png




In a way perhaps this would mean that only that one focus point is "real", while the other may be the effect of cartesian presentation of the form of the ellipse. Afterall, the sun does not move from one focal point to the other focal point:

http://ase.tufts.edu/cosmos/pictures...r1/Fig1_11.jpg

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Originally Posted by Kyriakos

@Mal4Mac: Is one of the two focal points of an Ellipse always in the same position as the axis of the cone the ellipse is a section of?

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Sorry, no. Here's a chat on physics forum about this matter:

http://www.physicsforums.com/showthread.php?t=277862

Don't ask me what a Dandelin sphere is!

I'd ask these kinds of technical questions on physics forum, and stick to more literary questions here.

It's interesting how these curves have come to describe certain kinds of argument in English - circular argument, one that assumes what it is trying to prove; elliptical argument, an argument that is (strictly) invalid because there is a missing premise. Example: All metals expand when heated, therefore iron will expand when heated. (Missing premise: iron is a metal). And of course there is hyperbole.

Have you found other literature on shapes? There's the Platonic dialogue where Socrates draws out a geometric proof from a slave boy, and E. A. Abbot's Flatland.

The famous sculptor Anish Kapoor has made use of many shapes, including the ellipse:

http://fabricarchitecturemag.com/art...sculpture.html

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Originally Posted by Kyriakos

The ellipse is one of the four types of conic sections, along with the parabola, the circle and the hyperbola. They are called conic sections because they are formed by the section of a (flat) surface onto a cylinder or a system of two cylinders united at their pinnacle points with that point being on the axis of every symmetry in their system...

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Did you mean cone rather than cylinder?

Yes, sorry :) (afterall a cylinder does not have a point as a pinnacle).

@MalMac: i am sad that the focal point is not in the axis of the cone... I will have to read more about that!

interesting thread Kyriakos. I will try myself and get acquainted with this complex mechanism/drawing concept and so on.
firstly what is the purpose of an elipse?
and
secondly as soon the circular is identified visually on the cone would you agree that aesthetically or schematically the cone does not longer appear as a cone but more of triangleish at the bottom and another kind of shape at the top?
in other word an elipse exaggerates the shapes of the cone making unconic or making drop its shape which means the ellipse is no longer an ellipse but just another circular shape.

Hi Cacian :)

I only now started getting interested in this issue too, but i think that the ellipse never really becomes a circle (for that to happen its focus points would have to be on the same point, and thus it would no longer be an ellipse). Maybe you are thinking of a cone as one which is formed by the rotation of a right angled-triange with the two sides of that angle being equal to each other. But this is only one type of cone. If the one side is vastly larger in lenght than the other one, then the cone would have a vast height in relation to the lenght of its base, and so any kind of ultra-thin ellipse can be formed as a section on it (up to the one which would tend to become a simple straight line, that is when the two focus points are on the periphery of the actual ellipse) :D

Btw, Mal4Mac, i am not sure if the answer given in your link is correct (or at least always correct). Have a look at these two images:


http://www.mathpages.com/home/kmath6...s/image001.gif

http://www.mathpages.com/home/kmath6...s/image002.gif

I am not certain if they are a special case of a cone, but at least in this case the focus point does indeed belong to the axis of the (real) cone the section was formed in :)

isn't there somewhere a camera obscura theorem waiting to come out?
i think the mistake here is drawing a straight line within an unstraight shape. one is to draw in parallel with shapes that are not of straight angles. straight lines is for a straight shape. parallel with uneven shapes. the reasons being that the ellipse or the sphere in the centre of the come is not even but tilted.
the same goes with the second picture. the sphere is tilted and therefore the lines are to be tilted too.
basically only draw a straight line going through a straight shapes .
a circle or a shape like a cone or titled rectangle will require a similar tilted lines if one needs to draw them otherwise it is uneven.

To me it seems to be like this:

The lines are not really straight, though, or rather they can be straight if you observe them from the top or the side in regards to the position of the actual cone in 3d space. Otherwise they are always tilted, but so is the rest of the shape, in the same way, which i think cancels out the need to present the lines as tilted in such a 2d figure of 3d space :) Those lines always are drawn as parallel to the axis of the cone, so however that is tilted, so are they.

ps: Do note that the lines which form the focus points on the 2d shape of the ellipse are indeed parallel to the conic axis, with one of them being the same as the conic axis, but only to the original cone where the section was made (not the cone or conic system of two cones which is produced by the section).