A simple question about conic properties...

[Kyriakos]

Hi, i would like to ask if someone can tell me whether one of the two focus points of an ellipse is always part of the axis of the cone that the ellipse was formed upon as its section by a plane/surface.

It seems from these two figures i found that it is true in at least specific cases, but i would like to ask if it is always true as long as the shape is an ellipse:

Focus points of an ellipse drawn as a surface:

Thanks for any helpful answer... I thought i could start the thread here, since the science forum has been dead for 4 days now so i might as well first try here...

 

[sanabas]

Nope.

I *think* that if either focus is on the cone's axis, then both are and the ellipse is actually a circle, but I may be wrong on that. *edit* Actually, I'm pretty sure this is the case. But I haven't bothered to prove it properly.*/edit*

Google 'Dandelin Spheres' for pretty pictures and more info.

Science forum isn't dead, it's just had no questions get asked.

 

[Kyriakos]

Hm, but if both focus points of an ellipse are on the conic axis, then they have to be both the very same point, not different points on the axis. If they are both the same point there there is only one focus in the ellipse, which by definition means it is not an ellipse but a circle.
By contrast, an ellipse pressuposes that there are two cones formed by the uneven section of the actual cone (as shown, again, in fugure 2) which is why it has two focus points (if it was a circle, then the two cones would be entirely symmetrical to each other in regards to their common base, which is not so in those which form an ellipse). Furthermore the fact that only one of those points is termed as "real" seems to signify it has some special characteristic in regards to the other one, which is termed as "empty". So if that special characteristic is not that it is part of the axis of the original cone, what is it?

(ps i will now go read about the dandelin spheres, but would like a reply to this post too if possible )

Edit: I now looked at the Dandelin spheres, here is an image of them:

I noticed, though, that in this image the conic section is in an inverted angle to the cone than the one in the images i posted before. This has to play a very significant role in regards to the system as a whole, so again i ask if in the other case of positioning of the section in regards to its angle to the base/pinnacle of the cone it remains true to say that one of the focus points will always be on the conic axis

 

[sanabas]

Hm, but if both focus points of an ellipse are on the conic axis, then they have to be both the very same point, not different points on the axis. If they are both the same point there there is only one focus in the ellipse, which by definition means it is not an ellipse but a circle.

A circle *is* an ellipse, in the same way that a square *is* a rctangle.

By contrast, an ellipse pressuposes that there are two cones formed by the uneven section of the actual cone (as shown, again, in fugure 2) which is why it has two focus points (if it was a circle, then the two cones would be entirely symmetrical to each other in regards to their common base, which is not so in those which form an ellipse).

I'm not following this. Where did you get those first two pics from? You construct/define an ellipse by intersecting a cone with a plane, rather than trying to construct two different cones around an ellipse (sort of, anyway). It's hard for me to tell if those cones are meant to have a circular base or not, and it looks to me like the point of the cone isn't in the centre of the cylinder the picture also shows. *edit* If you put the axis of that cone you draw to be parallel to the cylinder, then it won't be in the centre of the cylinder./you won't have a circular cone. If you connect the focus to the centre of the cylinder's base, and call that your axis, then you also won't have a circular cone, and the axis won't be parallel to the cylinder. I think you've just read too much into a non-accurate diagram, but I'm not sure without seeing more labels/what the diagram is trying to show.*/edit*

If you do try and reconstruct a circular cone above & below an ellipse like that, the two cones should be the same, just rotated 180 degrees, and the foci shouldn't be on the cones' axis (unless it's a circle).

Furthermore the fact that only one of those points is termed as "real" seems to signify it has some special characteristic in regards to the other one, which is termed as "empty". So if that special characteristic is not that it is part of the axis of the original cone, what is it?

Just something someone made up. It's not a 'real' focus, it's a 'full' focus. More to the point, it's a focus that is inside the sun, because of the huge difference in relative mass. Look at http://en.wikipedia.org/wiki/Gravitational_two-body_problem with two bodies of similar mass, both move in ellipses, and both have 2 'empty' foci. And if you made a model of just earth & sun, the sun would also move in an ellipse, it'd just be a very small ellipse, if you made a track of the centre of mass, it'd never be outside the rest of the sun, and you could say both foci are 'full'.

(ps i will now go read about the dandelin spheres, but would like a reply to this post too if possible )

Edit: I now looked at the Dandelin spheres, here is an image of them:

I noticed, though, that in this image the conic section is in an inverted angle to the cone than the one in the images i posted before. This has to play a very significant role in regards to the system as a whole, so again i ask if in the other case of positioning of the section in regards to its angle to the base/pinnacle of the cone it remains true to say that one of the focus points will always be on the conic axis

Neither of the focus points will be on the conic axis, unless the ellipse is a circle, in which case both will be. (but again, I haven't actually sat & proved it, I've only done enough to convince myself.)

If you want to visualise why, imagine you have that top sphere in the end of your cone. Now, turn the cone upside down. The very top of the sphere will be on the conic axis, yeah? Rest your plane inside. If it sits perfectly on top of the sphere, you'll get a circle. As you start to angle the plane in any direction, the one point where it touches the sphere will move, the further you tilt it, the further down the sphere the contact point moves, and the less circular the ellipse you get from where the plane & cone meet. Does that make sense? Now, that single point where the plane touches the circle (F1 in the diagram) is actually the focus of the ellipse made by that plane. (and if you then drop a much bigger sphere in, so it rests on top of the plane, the point where that one touches (F2 in the diagram) is the other focus) So the focus won't be on the conic axis.

As for the bit I bolded, yes it does. It's called the eccentricity of a conic section. You can basically think of it as the angle of the plane divided by the angle of the side of the cone. If the eccentricity is zero (meaning the plane is perfectly horizontal in your diagram) it makes a circle. If the eccentricity is less than 1, you get an ellipse. If the eccentricity is exactly 1 (meaning the plane is parallel to the side of the cone) you get a parabola. If the eccentricity is greater than 1, the plane will cut both the top & bottom sections of a double cone, and give you a hyperbola.

 

[Kyriakos]

I'm not following this. Where did you get those first two pics from? You construct/define an ellipse by intersecting a cone with a plane, rather than trying to construct two different cones around an ellipse (sort of, anyway). It's hard for me to tell if those cones are meant to have a circular base or not, and it looks to me like the point of the cone isn't in the centre of the cylinder the picture also shows. *edit* If you put the axis of that cone you draw to be parallel to the cylinder, then it won't be in the centre of the cylinder./you won't have a circular cone. If you connect the focus to the centre of the cylinder's base, and call that your axis, then you also won't have a circular cone, and the axis won't be parallel to the cylinder. I think you've just read too much into a non-accurate diagram, but I'm not sure without seeing more labels/what the diagram is trying to show.*/edit*

Hm, you are right that those two are not "cones" (not sure what they are called in English, or even in Greek...) since their base is not circular. However you also can note that the lines drawn from their pinnacle point are not their axis either, they are just vertical lines, at a right angle to the 2d surface the ellipse is drawn upon in the lower part of the image. So they still might serve so as to draw the focus points, and one of them seems to be exactly the same as the axis of the original (and real) cone the section was made upon.

I too do not have a proof that the "real" focus point of an ellipse, under the circumstance that it is drawn as the inverted one to the type of tilted ellipse always used in the Dandelin spheres illustration, is also a point in the axis of the cone the ellipse was created as a section of. It seems to me (instinctively) to be so, but i will surely look further into that. Thank you for your posts...

(ps: as to the origin of the images, check the image properties. I picked them up from a math site).

 

[Cheezy the Wiz]

Are you trying to understand how a CVT works?

Also, I initially read the thread title as "A simple question about colonic properties."

 

[Kyriakos]

Well, in a way i am looking for Colophonic properties (properties of the city of Colophon, which means 'Pinnacle' too anyway)

And no, i had not heard of CVT before. Is it some sort of practical use of conic properties? (it seems to be, from the images i saw in your link). I am interested in the conic shapes and currently on the form of the ellipse.

 

[Cheezy the Wiz]

Yes. What it does is, it by sliding the engine belt up and down two different cones (or rather, by moving the cones in and out), one can attain the different "gear ratios" smoothly without the jump that you get when you shift between two actual gears. It allows smooth acceleration. It also allows multiple engines to feed into a single drive train, such as in a hybrid vehicle.

 

[sanabas]

Hm, you are right that those two are not "cones" (not sure what they are called in English, or even in Greek...) since their base is not circular. However you also can note that the lines drawn from their pinnacle point are not their axis either, they are just vertical lines, at a right angle to the 2d surface the ellipse is drawn upon in the lower part of the image. So they still might serve so as to draw the focus points, and one of them seems to be exactly the same as the axis of the original (and real) cone the section was made upon.

The lines drawn from their pinnacle, perpendicular to the surface at the base, are their axis. And yes, one of them is therefore also identical to the axis of the original cone. But neither is a focus of the ellipse formed by the intersection of plane & cone, they are foci of what I took to be a circle, drawn at the very bottom of the diagram.

I too do not have a proof that the "real" focus point of an ellipse, under the circumstance that it is drawn as the inverted one to the type of tilted ellipse always used in the Dandelin spheres illustration, is also a point in the axis of the cone the ellipse was created as a section of. It seems to me (instinctively) to be so, but i will surely look further into that. Thank you for your posts...

There's no 'real' and 'empty' focus points. As I said, that's purely down to having a small object orbiting a very large object, the small object will move along an ellipse in which one focus is actually inside the large object. Make the two objects more similar in size, and it'll move along an ellipse in which both foci are 'empty'. And it's only a feature of a real-world orbit, you could define the same ellipse perfectly well without putting the sun & planet there.

(ps: as to the origin of the images, check the image properties. I picked them up from a math site).

http://www.mathpages.com/home/kmath631/kmath631.htm

OK, had a read of how they constructed that picture. The ellipse they're talking about is actually the bolder black line at the very bottom, and its two foci are the axes of those two cones. But they are not the foci of the ellipse being formed by the plane & cone intersecting in the first picture. Which I think is where you got confused by it. The circular cone that would give you the ellipse at the very bottom of the picture isn't shown anywhere, nor is its axis.

 

[sanabas]

Hi, i would like to ask if someone can tell me whether one of the two focus points of an ellipse is always part of the axis of the cone that the ellipse was formed upon as its section by a plane/surface.

The ellipse I have circled (well, ellipsed )in red is the one formed as the intersection of cone & plane. It's foci are not shown anywhere in the diagram. The one circled in green is perpendicular to the original cone's axis, and it is the one that has its focus on the original cone's axis. The cone that would intersect that horizontal plane to give that ellipse isn't shown anywhere in the diagram.

 

[Kyriakos]

Thank you again Sanabas

Although i have more questions now...

The plane is not in a right angle to the diameter of the base of the cone, so the ellipse drawn on it as a section would not have to have the shape/angle you posted (?). If the plane cut through the cone at a right angle to the diameter of the base (ie if the conic section was on a parallel 2d surface as the base of the cone) then that would be a special case, which indeed would make the section a circle. I do not see any other case where a conic section can be a circle at all. In the example given the cone obviously is not positioned in such a way to the plane. You seem to imagine that the "true cone" which would have that section created on it would exactly be one that would be placed in a way that the section could be only a circle.

***

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.

 

[sanabas]

Thank you again Sanabas

Although i have more questions now...

The plane is not in a right angle to the diameter of the base of the cone, so the ellipse drawn on it as a section would not have to have the shape/angle you posted (?). If the plane cut through the cone at a right angle to the diameter of the base (ie if the conic section was on a parallel 2d surface as the base of the cone) then that would be a special case, which indeed would make the section a circle. I do not see any other case where a conic section can be a circle at all. In the example given the cone obviously is not positioned in such a way to the plane. You seem to imagine that the "true cone" which would have that section created on it would exactly be one that would be placed in a way that the section could be only a circle.

Again, I'm not quite following you. You're correct that the only conic section that gives a circle would be by putting a plane through a cone, as with the red circled one, but perpendicular to the cone's axis.

If you have a plane intersect a cone, and the angle of the plane with the horizontal is less than the angle of the edge of the cone, you'll get an ellipse, just like the one circled in red. The two foci of that ellipse will be where those dandelin spheres touch the plane. Those foci are not shown in either of your first two pictures.

If you take that red-circled ellipse, project it down onto a horizontal surface which is perpendicular to the cone's axis, you'll get an entirely different ellipse, just like the one circled in green. The two foci of that ellipse will be the two axes of the cones shown in your 2nd picture. The ellipse itself will not match the edges of your original cone, they will be inside it.

If you wanted to, you could take the measurements of that green-circled ellipse, and work out the shape/angle of the cone that would intersect that horizontal plane to give it to you. That cone is also not shown anywhere on your diagram, and if you constructed it, the axis would not be up & down, would not be perpendicular to that horizontal plane, would not be parallel to the axis of your original cone. It would actually be off at an angle. Like this:

Excuse how crap that looks, I've only got paint. Red line is the cone's axis. 2 foci can be seen on the diagram still. They will not match.

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.

No. I'm still not sure what you mean by 'true' focus point. Neither focus of the red-circled ellipse will be on the axis of the cone, except for the special case of it being a circle. The pictures you find of Dandelin spheres show where the foci are, they are at the one & only point where each sphere meets the plane. One sphere above, one below, to give the 2 different foci.

 

[Kyriakos]

Excuse how crap that looks, I've only got paint. Red line is the cone's axis. 2 foci can be seen on the diagram still. They will not match.

Thank you yet again... but now i think there is an error in what you presented (unless my own view is heavily erroneus...) let me explain:

In your image you placed the actual ellipse above the basis of the cone (which basis is not even shown), with the ellipse being a non-tilted one since the cone itself is tilted now. By doing this you reversed the positions of the ellipse and the basis in regards to how the axis of the cone is related to the focus points. This meant that the axis of the cone in your example has been moved to the side, which seems also obvious from the fact that you have the axis of the cone going through exactly the point where the minor and major diameters of the ellipse meet, rendering your ellipse a circle by definition, which cannot be true if the premise you based your image upon was also true.

Of course the ellipse could have had its major diameter towards the other side, but then the cone would not be at all drawn in the manner of your image, since then it would be drawn as one whose axis would appear to be (in a 2d shape of the same perspective you used) vertical to the plane which caused the section.

 

[Leoreth]

It's very hard to parse what you're saying but I don't think your objections make sense.

First of all, whether the cone or ellipsis is tilted is irrelevant. What matters is if they still meet the definition of cone and ellipsis, which they do. Your objection that the ellipsis would be a circle in this situation (even though I'm not sure how you arrived at this conclusion) doesn't matter as well, since all circles are ellipses.

(This is unrelated to the argument but afaik cones don't need to have a circular base, so the base of this particluar cone (which is identical to its intersection with the plane) could possibly be a non-circular ellipsis already. We can't tell because of the perspective.)

Sanabas constructed a valid example of a cone and an ellipsis here in which your claim does not hold. Looks fine to me.

 

[Kyriakos]

It's very hard to parse what you're saying but I don't think your objections make sense.

First of all, whether the cone or ellipsis is tilted is irrelevant. What matters is if they still meet the definition of cone and ellipsis, which they do. Your objection that the ellipsis would be a circle in this situation (even though I'm not sure how you arrived at this conclusion) doesn't matter as well, since all circles are ellipses.

(This is unrelated to the argument but afaik cones don't need to have a circular base, so the base of this particluar cone (which is identical to its intersection with the plane) could possibly be a non-circular ellipsis already. We can't tell because of the perspective.)

Sanabas constructed a valid example of a cone and an ellipsis here in which your claim does not hold. Looks fine to me.

In common usage in elementary geometry, cones are assumed to be right circular, where right means that the axis passes through the centre of the base (suitably defined) at right angles to its plane, and circular means that the base is a circle. Contrasted with right cones are oblique cones, in which the axis does not pass perpendicularly through the centre of the base.[1] In general, however, the base may be any shape that permits a circular cross section of the cone, and the apex may lie anywhere (though it is usually assumed that the base is bounded and therefore has finite area, and that the apex lies outside the plane of the base).

When people do not otherwise define a "cone" it is obviously meant as a right-circular cone (ie one formed by the full rotation of a right-angled triangle, with its basis being the (circular) periphery which has as a radius one of those sides of the triangle that are vertical to each other. Furthermore most of the time it is taken for granted that the triangle in question does not have those sides being of equal length.

So thank you for your good input there Leoreth. I suppose you could have tried to guess that no one here was trying to examine the properties of the axis of the cone in relation to its focus points without at least having as a non-variable the type of form the basis of the cone here has, ie a circle. So if you have any good info on that, please share, otherwise it's been fun i guess.

 

[Leoreth]

Well, I'm sorry that I misunderstood you, but I suppose you have only yourself to blame for that since you're really communicating yourself very badly. I haven't really been doing any "elementary geometry" for a long while so I approached your assertions from the most general point of view possible. Because you were, even if not intentionally, making a very general statement. It's not my job to read your mind and figure out what further assumptions you've made because they seem appropriate to you. That is not how math works.

So I suppose what this comes down to is once again this:

Spoiler :

 

[Kyriakos]

Well, I'm sorry that I misunderstood you, but I suppose you have only yourself to blame for that since you're really communicating yourself very badly. I haven't really been doing any "elementary geometry" for a long while so I approached your assertions from the most general point of view possible. Because you were, even if not intentionally, making a very general statement. It's not my job to read your mind and figure out what further assumptions you've made because they seem appropriate to you. That is not how math works.

Ok Leoreth, I am now convinced you are at least some sort of mathematical genius by whom it is regarded as something elementary to prove whether or not the conic axis of a right circular based cone is under some circumstances- as those mentioned above- including the one of the two focus points of the ellipse formed as a section of that cone.

Funny, though, if that is so elementary to you, where exactly is the proof that the above is true or false?



Edit: In some posts above i mentioned that it seemed to me that in the ellipse (which i meant as one sectioned on a right cone with a circular base) indeed the axis of the cone has one of the focus points of the ellipse as part of it. I still would love to read if this is indeed true in the case that the ellipse is a right circular-based one, given that at the moment that is all i am interested in (i am not using a different kind of ellipse in my current research).

So thanks again to any who want to help with this

 

[Goodfella]

Dude it was your wikipedia article called that called it "elementary geometry" in the first place.

 

[Rashiminos]

I would think that it only applies when the ellipse in question is also a circle (the axis of the right cone passes through the center of a circle, an ellipse that is also a circle has both foci at the center of the ellipse).

 

[Kyriakos]

Hm, thank you all again...

I did some more thinking, and i am of the view that maybe what i was calculating up to now was not if the focus points (on the 2d shape of the ellipse) are points in the line of the conic axis, but if the point where the conic axis cuts throught the actual 3d-environment shape of the ellipse is in some constant relation to the focus points... This would allow one (perhaps) to calculate where the point the conic axis cuts through the 3d ellipse is placed, if he knows the position of either both of the (2d) focus points, or the one of them which is on the same "side" as the axis point in the ellipse...

So help with that would be really welcome, and sorry for the false understanding of what i was looking for! :/