Conic sections-The ellipse

[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 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/2011/01/conic-section-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.

[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.

[Kyriakos]

I surely will...Thanks :)

[cafolini]

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.

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.

[mal4mac]

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.
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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.virginia.edu/lectures/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.

[Kyriakos]

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


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.

[cafolini]

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.

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

[Kyriakos]

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/wikipedia/commons/thumb/6/65/Ellipse_Properties_of_Directrix_and_String_Constru ction.svg/278px-Ellipse_Properties_of_Directrix_and_String_Constru ction.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/Explore_figs_5/Chapter1/Fig1_11.jpg

[mal4mac]

@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?

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/articles/0110_sk_sculpture.html

[Nick Capozzoli]

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...

Did you mean cone rather than cylinder?

[Kyriakos]

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!

[cacian]

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.

[Kyriakos]

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/kmath631/kmath631_files/image001.gif

http://www.mathpages.com/home/kmath631/kmath631_files/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 :)

[cacian]

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.

[Kyriakos]

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).

[mal4mac]

Btw, Mal4Mac, i am not sure if the answer given in your link is correct...


I'd expect there might be one slice where the focus coincides with the axis. But look at this:

http://www.clowder.net/hop/Dandelin/ellhyp.gif

It seems quite obvious the axis is no way near the focus. Check out this for some more great pictures of Dandelin spheres:

http://www.clowder.net/hop/Dandelin/Dandelin.html

I especially like this one:

http://www.clowder.net/hop/Dandelin/Ocean.gif

A floating ball head [Dandelin sphere] wearing a dunce-cap
The ocean surface plane defined by the cone meeting the water is the conic section (here it's an ellipse).
Where the ball head touches the water is a focus.
Where the fish [Dandelin sphere] kisses the air is a focus.

Note, I've simplified the original description. For instance, I don't see why a mosquito net is needed!

Wonderful resource on Japanese temple geometry (they were obsessed with ellipses):

http://www.physics.princeton.edu/~trothman/templegeometry.pdf

[cacian]

Hi Kyriakos
what i am trying to say is that in the second picture for example. the lines going through the axis are straight but the ellipse or axis is not even on the surface it being drawn. it is tilted and therefore the ellipse will appear uneven compare to the top one on the cone. the reason for this is that one is trying to force a straight line from top to bottom going through the ellipse which is tilted. the end result is the axis appear uneven or uncentered.
therefore the theory is flawed or shall i say it does not work
it is not the ellipse that the issue here it is the line that is drawn straight. in other one cannot draw a straight line through the ellipse and expect evenness. it does not work.
so either draw a tilted line to go according to the ellipse or don't either way it is not going to work because the top circle of the cone is going to come uncentered too.
so there is no such a thing uncentered ellipse it is the drawing of the line that is uncentered or the issue.

[Kyriakos]

Thank you both for your replies :D

I think (not sure) that this is what could be said here:

@Cacian: the ellipse is not uncentered, if it is drawn as a 2d shape. Then its obvious "center" is the point where its major and minor axis meet. However in a 3d space the ellipse is not drawn on a surface, but has two very uneven in size curves. So the focus points are related to those curves in 3d space, in fact they seem to be the "center" of the smaller curve (in reality the smaller cone made as part of a conic system after the original larger cone, as shown in my figure #2), and then the point in equal distance to it from the other side of the 2d ellipse "center" point.

@Mal4Mac: from the images i saw of the Dandelin spheres they always are formed in the inverted ellipse in regards to the angle of the ellipse towards the base of the cone. In my own image the cone is inverted. In your own it is upright (what matters is not the cone by itself, but also the angle of the ellipse in 3d space). So it seems to me that as long as the ellipse is angled/tilted in the way i presented in my image, it always will have its real focus point in the axis of the cone. Whereas if it is angled in the inverted way, the axis will not have any focus point in it...

[cacian]

However in a 3d space the ellipse is not drawn on a surface, but has two very uneven in size curves.
exactly and that is because the straight line is being forced/ going through it. take the line out and the curves are perfectly even without interrupting it with a line to measure it.
one does not measure using a diagram it does not work:)

[mal4mac]

@Mal4Mac: from the images i saw of the Dandelin spheres they always are formed in the inverted ellipse in regards to the angle of the ellipse towards the base of the cone. In my own image the cone is inverted. In your own it is upright (what matters is not the cone by itself, but also the angle of the ellipse in 3d space). So it seems to me that as long as the ellipse is angled/tilted in the way i presented in my image, it always will have its real focus point in the axis of the cone. Whereas if it is angled in the inverted way, the axis will not have any focus point in it...

Sorry, I don't see that at all. Can you prove it, or produce a few diagrams that might convince me? Or find a page that supports your statement?

[Kyriakos]

I cannot prove it, although to me (again instinctively) it seems to be true. I will certainly look a lot more into this though... At any rate if the ellipse is tilted in the manner/direction shown in the Dandelin illustration, it is obvious it won't ever have a focus point in the axis of the cone.

[Kyriakos]

Hm... I did some more thinking/calculations, and i wonder if the true focus point of the ellipse is the point where the axis of the cone meets the major axis of the ellipse, as long as that point they meet in is in the periphery which is smaller than the base of the cone and is further away from the basis of the cone than it is to the pinnacle of the cone.

Might be true. Will rework on it though.

[Kyriakos]

Did some more reading on this, changing my view :)

http://upload.wikimedia.org/wikipedia/commons/thumb/7/76/Ellipse_parameters_2.svg/400px-Ellipse_parameters_2.svg.png

What i was looking for was not, in the end, the actual position of the conic axis as one identified in the corresponding focal point on the side of the cone where the smaller of the curved parts of it is( since indeed i now know this would mean the ellipse is a circle), but for the relative distance of the conic axis to that focus point, as opposed to the distance of the conic axis to the center of the ellipse :)

I think that the opposite focus point will always be in a larger linking line to the Latus Rectum (which is the chord uniting the edges of the smaller curve of the ellipse), or at least its center, than the focus on its own side. Often it would be a case of one of the straight lines of a right triangle, and the hypothenuse linking them. I wonder if this always will be the case as a ratio between the two distances (given a specific example of such a triangle and its sides). My will was just to know under which minimal requirements involving the conic axis and the focus points, one can tell what a particular element related to the eccentricity of the ellipse will be: how it is tilted in regards to the base of the right/circular based cone.

In other words, what i am asking is if you can know which vertex of the ellipse is the higher one, if you define height not in regards to the cone itself (cause it can be inverted) but to a cartesian, stable axis.

[YesNo]

In other words, what i am asking is if you can know which vertex of the ellipse is the higher one, if you define height not in regards to the cone itself (cause it can be inverted) but to a cartesian, stable axis.

I never heard of Dandelin spheres before and the thread is interesting.

Perhaps this is completely incorrect, but it seems to me that there are infinitely many planes that would cut a cone to project the same ellipse (or other conic section) onto a 2 dimensional space. Looking at it from the perspective of a particular plane that cuts the cone, one of the foci would be further away from that axis of the cone, however, once that set of points in the intersection of the cone with the plane is mapped to a 2 dimension space, that information is lost.

Just a guess based on the previous discussion.

[Kyriakos]

You are correct :) However the actual ellipse might be said (although it doesn't have to, and it seems in reality it does not either) to exist in a specific 3d location, unlike the 2d projection of it which is mostly some sort of standard for an unlimited number of different cases of surface tilt or cone tilt or cone size...

However by now i gathered that one can never know the actual tilt of the ellipse if he only knows information relative to the conic axis, the focus points, and that the shape is indeed an ellipse (even if he also knows that its minor axis is a lot smaller than the major axis). This has to do with the 3d shape being not bound by any stable direction that the 2d shape- or we as humans- are, eg the set downward direction in the cartesian axis, or the force of gravity which pulls one down, with "down" being the same for a relatively small part of the environment he is in (ie he is not in the antipodal parts of the planet, or something significantly towards that).

However i needed to examine this for a short story, which i now completed, and in the story obviously gravity is involved, so "down" is indeed a set direction for the narrator. Moreover the impossibility to know a tilt for the 3d shape of the ellipse is incorporated in the storyline now as well :)