A simple question about conic properties...

[peter grimes]

The axis of the cone will only pass through the foci of an ellipse in the special case of a circle, as others have already said. I'm not a mathematician so I'm not going to attempt a proof, though I suspect it's a fairly straightforward one.


If you doubt this then try hand drafting a few ellipses of varying eccentricities. In fact - you should do this anyway. Don't take our words for it

EDIT: Another way to state this is "The axis of the cone will always pass through the intersection of the major and minor axes of the ellipse."

 

[Kyriakos]

Hm:

:/

I guess i can live with shifting focus (bad pun) onto the chord formed between the identical bow-sections of the ellipse, and its relation to the focus point. According to wiki it is parallel to the directrix, and the focus point splits it in the center when viewed from the top angle.
Given that the position of this chord gives away the actual degree and type of tilted ellipse there exists, i suppose i can make do without more interest on the conic axis itself.

(way too much work to construct a single system of metaphor there for a story) :/

 

[sanabas]

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

It is.

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.

Yes, which was my point. If paint let me rotate the image by an angle that's not a multiple of 90 degrees, I'd have done that so that it did conform to the convention of upright cone and tilted ellipse. Just turn your head so it looks that way. (p.s. there is no such thing as a 'tilted'/'non-tilted' ellipse).

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,

Just a fluke, I tried to bisect the apex of the cone by eyeball, and that's where the line ended. The cone wasn't very accurate in the first place. It's just a paint diagram to illustrate what I'm talking about. You can not do geometric proofs just by looking at a not particularly accurate diagram.

rendering your ellipse a circle by definition, which cannot be true if the premise you based your image upon was also true.

OK, now you're really confusing me. You're saying that if the axis of a cone passes through the centre of an ellipse, that ellipse must be a circle? Why?
Is that because you're also saying that the axis of a cone must pass through the focus of an ellipse formed using that cone? And so if centre of ellipse = focus of ellipse, it's a circle?

Again, the fact the axis is *close* to the centre is just the way I drew it. It's an illustration, not an accurate diagram.

Let me have another go, I've taken a screenie, rotated it, pasted it, and done my best to make an accurate cone around the original, green-circled ellipse we were talking about. I've also extended the original axis we were talking about, the one that does intersect a focus. As you can see, the new axis is not parallel to the original axis. As you can see, the new axis does not intersect a focus.

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.

I have done enough to convince myself of this. The only circumstances in which one of the foci of an ellipse formed as a section of a right circular cone lies on the axis of that cone is when both foci do, and the ellipse is a circle.

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

I put it in this thread already:

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.

If there's anything there that looks obviously wrong, or isn't making sense, please tell me.

If you want a proof of why the sphere only touches the ellipse at one point, and why that point is a focus, you can find formal proofs by doing more reading on dandelin spheres.

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...

No, it's not. I think it will be on the major axis, between the foci, and related to the eccentricity. The lower the eccentricity, the closer to the centre it will be. The greater the eccentricity, the closer to a focus. If the eccentricity is zero, it will actually be in the centre. (So I think you were right earlier, if the conic axis intersects both major & minor axes of the ellipse, then it is a circle). I have a feeling that distance from centre of ellipse to conic axis divided by distance from centre to focus will equal distance between foci divided by length of major axis, which equals the eccentricity. But that is just intuition, I could be very wrong. And it just looks like a curiosity, not a useful result.

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...

Yes, if you have both foci, and the eccentricity of the ellipse, you can reconstruct a cone & plane, and so see where the conic axis would cut through the ellipse.

 

[sanabas]

EDIT: Another way to state this is "The axis of the cone will always pass through the intersection of the major and minor axes of the ellipse."

Major axis, yes. Minor axis, no. Imagine a very eccentric ellipse in your cone, tilted to almost the exact same angle as the side of the cone itself. One end of the major axis will be almost right at the vertex of the cone, and the axis of the cone will obviously pass through the ellipse very close to that end. But the minor axis will be much further away.

 

[sanabas]

Hm:

:/

I guess i can live with shifting focus (bad pun) onto the chord formed between the identical bow-sections of the ellipse, and its relation to the focus point. According to wiki it is parallel to the directrix, and the focus point splits it in the center when viewed from the top angle.
Given that the position of this chord gives away the actual degree and type of tilted ellipse there exists, i suppose i can make do without more interest on the conic axis itself.

(way too much work to construct a single system of metaphor there for a story) :/

You've lost me again with 'bow-section'. You're talking about the latus rectum?

Simply having that doesn't give you enough info to define an ellipse. It tells you nothing about the eccentricity. You can have multiple, different ellipses with an identical latus rectum. To specify a single one, you'd need to know the eccentricity, know the distance from the focus to the directrix, or know the co-ordinates of the focus and the equation of the directrix.

 

[Rashiminos]

Nevermind.

Major axis, yes. Minor axis, no. Imagine a very eccentric ellipse in your cone, tilted to almost the exact same angle as the side of the cone itself. One end of the major axis will be almost right at the vertex of the cone, and the axis of the cone will obviously pass through the ellipse very close to that end. But the minor axis will be much further away.

It probably doesn't even have to be that close (I now think having an ellipse with nonzero eccentricity would imply the conic axis not intersecting the minor axis, but I haven't completely worked it out).

A rough picture of the cone case:


The black lines represent a triangular cross-section of the cone. The blue line is a circle viewed from the side. The red line is an ellipse viewed from the side. The small black boxes show where the endpoints of the major axis of the ellipse would have to be in order for the conic axis to intersect the center of the ellipse. The hash mark is an approximation of where the center of the ellipse would be.

EDIT: Another way to state this is "The axis of the cone will always pass through the intersection of the major and minor axes of the ellipse."

OTOH peter, a plane intersecting a cylinder to form an ellipse will always have that property.

 

[peter grimes]

Major axis, yes. Minor axis, no. Imagine a very eccentric ellipse in your cone, tilted to almost the exact same angle as the side of the cone itself. One end of the major axis will be almost right at the vertex of the cone, and the axis of the cone will obviously pass through the ellipse very close to that end. But the minor axis will be much further away.

OTOH peter, a plane intersecting a cylinder to form an ellipse will always have that property.

Yes, I see that I was wrong - thanks

Ellipses are a common issue I encounter at work, and I've developed a few favorite ways of generating them. It's always necessary for me to establish the intersection of the major and minor axes, so I mistakenly imagined it to have more importance than it does.

 

[sanabas]

It probably doesn't even have to be that close (I now think having an ellipse with nonzero eccentricity would imply the conic axis not intersecting the minor axis, but I haven't completely worked it out).

Yeah, it's just easier to visualise with a high eccentricity. I agree with you, and I posted about it a few posts ago:

I think (the cone's axis) will be on the major axis, between the foci, and related to the eccentricity. The lower the eccentricity, the closer to the centre it will be. The greater the eccentricity, the closer to a focus. If the eccentricity is zero, it will actually be in the centre. (So I think you were right earlier, if the conic axis intersects both major & minor axes of the ellipse, then it is a circle). I have a feeling that distance from centre of ellipse to conic axis divided by distance from centre to focus will equal distance between foci divided by length of major axis, which equals the eccentricity. But that is just intuition, I could be very wrong. And it just looks like a curiosity, not a useful result.

 

[Kyriakos]

Thanks again

Some (i hope) final questions:

-If one knows roughly, but not entirely where the two focus points are on the major axis of the ellipse, and knows which of the two is connected vertically to the chord formed as a border between the smaller and the larger curves of the ellipse (the chord being called Latus Rectum) but without being able to actually measure the distance between that focus point and the chord, nor to be sure if the cone is inverted or upright, can he then at least determine the type of the eccentricity of the ellipse in regards to which way it is tilted if he also has a rough idea of the relative position of the conic axis to the focus points?

Mostly going on by this:

Each focus F of the ellipse is associated with a line parallel to the minor axis called a directrix. Refer to the illustration on the right, in which the ellipse is centered at the origin. The distance from any point P on the ellipse to the focus F is a constant fraction of that point's perpendicular distance to the directrix, resulting in the equality e = PF/PD. The ratio of these two distances is the eccentricity of the ellipse. This property (which can be proved using the Dandelin spheres) can be taken as another definition of the ellipse.
Besides the well-known ratio e = f/a, where f is the distance from the center to the focus and a is the distance from the center to a vertex (most sharply curved point of the ellipse), it is also true that e = a/d, where d is the distance from the center to the directrix.

Basically what i am trying to present in the literary work is that one has only a view of the 2d form of the ellipse, and is trying to realize which way to go in its periphery so as to arrive at a higher point, and in the end to the highest vertex (point connecting the identical bows of the ellipse), for which he should at least know if the cone is inverted or not and which focus point is the one vertically cutting through the Latus Rectum.

It is mostly just an allegory for lost orientation.

I want to thank you for all your help up to now, and i am sorry for the (many...) misunderstandings, which were caused on the one hand by the fact that i just now returned to observing this geometrical phenomenon and its related properties, and on the other the fact that i was not familiar with the english terms for the various parts being identified...

Later edit:

Also:

I think (the cone's axis) will be on the major axis, between the foci, and related to the eccentricity. The lower the eccentricity, the closer to the centre it will be. The greater the eccentricity, the closer to a focus. If the eccentricity is zero, it will actually be in the centre. (So I think you were right earlier, if the conic axis intersects both major & minor axes of the ellipse, then it is a circle). I have a feeling that distance from centre of ellipse to conic axis divided by distance from centre to focus will equal distance between foci divided by length of major axis, which equals the eccentricity. But that is just intuition, I could be very wrong. And it just looks like a curiosity, not a useful result.

Is it true to say that the conic axis in the case we have an ellipse (not a circle or line) will always be closer to the focus point that has the smaller distance between it and the middle of the Latus Rectum? Moreover, if that is indeed true, does the exact middle of the latus rectum have to be mentioned, and not just any point in it?

*I think that the opposite focus point will always be in a larger linking line to the Latus Rectum 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.

 

[sanabas]

Thanks again

Some (i hope) final questions:

-If one knows roughly, but not entirely where the two focus points are on the major axis of the ellipse, and knows which of the two is connected vertically to the chord formed as a border between the smaller and the larger curves of the ellipse (the chord being called Latus Rectum) but without being able to actually measure the distance between that focus point and the chord, nor to be sure if the cone is inverted or upright, can he then at least determine the type of the eccentricity of the ellipse in regards to which way it is tilted if he also has a rough idea of the relative position of the conic axis to the focus points?

I can't work out what you want, sorry. I don't know what you mean by 'connected vertically'. I don't know what you mean by 'inverted or upright'. I don't know what you mean by 'which way it is tilted.'

The latus rectum, by definition, goes through a focus. It's not a border between two curves, it's a line throgh a focus, parallel to a directrix, perpendicular to the major axis. There's zero distance between focus and that chord. The only difference between it and any other chord perpendicular to the major axis is that it goes through a focus.

I *think* what you're getting at is something like a building/ship in the shape of a cone. With a floor within that building, at an angle to the building, and so the shape of the floor is an ellipse. With the cone's axis being something like a lift shaft, and the means of escaping up or down within the building. And for someone on that floor to work out for themselves where that axis is, by examining the floor. And to work out if they're oriented with the cone, as though it's a pyramid, and they can go up to the vertex, or the other way, like an ice-cream cone, and they need to go down to the vertex. Is that close?

If it is, then you can work out the eccentricity simply by working out the length of the major & minor axes, you don't need to know anything about the cone's axis. Once you have the eccentricity, you can work out the foci. Once you have the eccentricity & foci, you can (I think) get two possible locations for the cone's axis. You can also work out where the directrix and latus rectum are. However, there is no way to determine whether the vertex of the cone is up or down, and there is no way to work out the tilt of the floor unless you already know the tilt of the cone's side.

Basically what i am trying to present in the literary work is that one has only a view of the 2d form of the ellipse, and is trying to realize which way to go in its periphery so as to arrive at a higher point, and in the end to the highest vertex (point connecting the identical bows of the ellipse), for which he should at least know if the cone is inverted or not and which focus point is the one vertically cutting through the Latus Rectum.

No, there is no way to work out which is the 'highest' vertex. It's like being in an equilateral triangle and trying to work out which of the three corners is the 'top'.

It's not 'the' latus rectum. Each focus has a latus rectum. Each focus has a directrix.

It is mostly just an allegory for lost orientation.

You've successfully lost me a couple of times along the way.

Is it true to say that the conic axis in the case we have an ellipse (not a circle or line) will always be closer to the focus point that has the smaller distance between it and the middle of the Latus Rectum? Moreover, if that is indeed true, does the exact middle of the latus rectum have to be mentioned, and not just any point in it?

No. There is more than one latus rectum. The exact middle of a latus rectum *is* a focus.

*I think that the opposite focus point will always be in a larger linking line to the Latus Rectum 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.

Again, for the last bit, there's no way to work out how it is tilted, which end is 'up' and which end is 'down'. There is no way to work out the angle of the tilt, unless you also know the angle of the side of the cone. You can work out eccentricity purely from the length of the major & minor axes, and the tilt of the ellipse divided by the tilt of the cone's edges also equals the eccentricity.

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.

No. As I said, it's like working out which corner of a triangle is the top one.

 

[Kyriakos]

I *think* what you're getting at is something like a building/ship in the shape of a cone. With a floor within that building, at an angle to the building, and so the shape of the floor is an ellipse. With the cone's axis being something like a lift shaft, and the means of escaping up or down within the building. And for someone on that floor to work out for themselves where that axis is, by examining the floor. And to work out if they're oriented with the cone, as though it's a pyramid, and they can go up to the vertex, or the other way, like an ice-cream cone, and they need to go down to the vertex. Is that close?

If it is, then you can work out the eccentricity simply by working out the length of the major & minor axes, you don't need to know anything about the cone's axis. Once you have the eccentricity, you can work out the foci. Once you have the eccentricity & foci, you can (I think) get two possible locations for the cone's axis. You can also work out where the directrix and latus rectum are. However, there is no way to determine whether the vertex of the cone is up or down, and there is no way to work out the tilt of the floor unless you already know the tilt of the cone's side.

Thank you so much, Sanabas

Yes, your own example/guess is pretty close to what is going on in my short story, although the end there is not to reach the actual axis of the cone since only the periphery is accessible (as if one is a planet moving in that elliptical periphery). The end is just to be able to know which way to go so as to reach the point in the periphery which is at the same time the connecting dot of the two smaller and identical parts of the ellipse (which is our point of confusion, i will refer to it in the following lines...) and also the higher part of it in relation to a fixed axis (like a certesian axis). I think the confusion here is mostly derived from my use of the chord linking the two smaller bows of the ellipse, which i view as something in space (away from the focus point which i view as something set primarily in the 2d form of the ellipse) and you use (probably more correctly) as just a parallel line to the smaller axis of the ellipse, ie again something set in the 2d image of the ellipse.

I gather that for one to reach a point in the periphery of this ellipse which would be on its higher position in regards to a fixed cartesian axis, and also to be formed between the two smaller identical bows of the ellipse, he would always have to be moving in an upright cone and never an upside-down cone (do note that i allowed only for those two cases in my story, either entirely upright cone, or entirely upside down, resting on its inverted pinnacle).

The story mostly, afterall, is the wish of the narrator that he is already moving near the smaller bit of the ellipse (supposing the latus rectum cuts up the ellipse into two uneven curves) because on the one hand he wants to rise up, on the other he wants to think that he won't have to move for an eternity, which he might have had if he now is nearer the lower vertex (ie the one nearer the base of the cone).

Again thank you very much. Of course i would be utterly grateful (or even moreso) if you can also reflect on this new, hopefully final post of mine

 

[sanabas]

Thank you so much, Sanabas

Yes, your own example/guess is pretty close to what is going on in my short story, although the end there is not to reach the actual axis of the cone since only the periphery is accessible (as if one is a planet moving in that elliptical periphery). The end is just to be able to know which way to go so as to reach the point in the periphery which is at the same time the connecting dot of the two smaller and identical parts of the ellipse (which is our point of confusion, i will refer to it in the following lines...) and also the higher part of it in relation to a fixed axis (like a certesian axis). I think the confusion here is mostly derived from my use of the chord linking the two smaller bows of the ellipse, which i view as something in space (away from the focus point which i view as something set primarily in the 2d form of the ellipse) and you use (probably more correctly) as just a parallel line to the smaller axis of the ellipse, ie again something set in the 2d image of the ellipse.

I gather that for one to reach a point in the periphery of this ellipse which would be on its higher position in regards to a fixed cartesian axis, and also to be formed between the two smaller identical bows of the ellipse, he would always have to be moving in an upright cone and never an upside-down cone (do note that i allowed only for those two cases in my story, either entirely upright cone, or entirely upside down, resting on its inverted pinnacle).

The story mostly, afterall, is the wish of the narrator that he is already moving near the smaller bit of the ellipse (supposing the latus rectum cuts up the ellipse into two uneven curves) because on the one hand he wants to rise up, on the other he wants to think that he won't have to move for an eternity, which he might have had if he now is nearer the lower vertex (ie the one nearer the base of the cone).

Again thank you very much. Of course i would be utterly grateful (or even moreso) if you can also reflect on this new, hopefully final post of mine

OK. So, have we cut the ellipse into two parts by using the actual latus rectum, which goes through the focus? Or simply by using a chord parallel to the minor axis, which may or may not be close to a focus? I still can't tell, and don't know why the focus is relevant anyway.

For the bolded bit, you gather wrongly. As shown in your very first post, in the 2nd picture, you will get an identical ellipse regardless of which way up the cone is.

To know the highest point in relation to a fixed cartesian axis, you have to define where that fixed cartesian axis actually is. Where is the origin? Which direction does the positive x axis point? etc.

What you're asking is basically the equivalent of saying that the date today is June 25th, 2013. If we cut the earth's orbit in two with a latus rectum through the 'empty' focus, then the earth's position is somewhere in the smaller bit of its elliptical orbit. It will reach the very end point of that elliptical orbit on July 5th 2013. With respect to a fixed cartesian axis, or with respect to the rest of the universe, are we moving higher or lower over the course of those 10 days? Is the cone that would give us the earth's orbit pointing up or down? They're nonsensical questions. 'Up' & 'down' are just arbitrary labels. Is the north pole the top or bottom of the earth, or neither? Right now, is the moon above or below the earth, and by how much? Are we above or below the sun? Which number is printed on the top of 6-sided dice? The narrator wanting to rise up is the same sort of thing, there is no up, it's just an arbitrary label for a particular direction.

 

[Kyriakos]

In the purely geometrical world, yes, but not in a world where you have gravity acting in a limited environment and so having always (seemingly, but in reality for that small environment) one direction as a force. The narrator is human, so gravity itself dictates which way is up. Without gravity you do not have any intuitive (or other) reason to name as "up" that which is on the northern part of the cartesian axis as they are usualy drawn.

So, since the narrator is only trying to define "up" in regards to his own location, and not some more significant part in the universe or even (at least) the whole of the earth, the definition is warranted.

Which is pretty much the point in the story, given that only if one is at the highest point will he then be able to re-start his spin on the periphery (which in the setting of the story would be sustainable under some circumstances) by allowing himself to fall down from that.

edit: as for your point about the identical ellipse in two cones which are set one as upright, the other as inverted, 180degrees of difference: i guess you mean that the second ellipse would be the same as the first one, but that the second cone will surely be of a quite different size than the first one? This does not matter much in the setting of my own story, given that, again, the main attempt is to reach the highest point (relative to the lowest being one connected to gravity, ie more to the south part of the cartesian axis). The size of the two possible cones is regarded as equal at all times, so in effect you cannot have the same ellipse in one cone and a different sized inverted one. Of course, for reasons described, the narrator would like to believe that the highest point is indeed the one closer to the latus rectum, for reasons of practicality (less distance to travel to).

 

[sanabas]

In the purely geometrical world, yes, but not in a world where you have gravity acting in a limited environment and so having always (seemingly, but in reality for that small environment) one direction as a force. The narrator is human, so gravity itself dictates which way is up. Without gravity you do not have any intuitive (or other) reason to name as "up" that which is on the northern part of the cartesian axis as they are usualy drawn.

So, since the narrator is only trying to define "up" in regards to his own location, and not some more significant part in the universe or even (at least) the whole of the earth, the definition is warranted.

OK, then it's easy. Walk uphill, you reach the highest point. It's easy to tell which way uphill is, just pour some water on the floor, see which way it flows. If straight down according to gravity is actually perpendicular to the floor, the way it normally is in houses, then we're back where we started, that neither end can be considered 'up'.

edit: as for your point about the identical ellipse in two cones which are set one as upright, the other as inverted, 180degrees of difference: i guess you mean that the second ellipse would be the same as the first one, but that the second cone will surely be of a quite different size than the first one?

No, I mean that the cones will be identical. Same size, same shape, just rotated 180 degrees. Just as it is shown in the 2nd picture of your first post.

This does not matter much in the setting of my own story, given that, again, the main attempt is to reach the highest point (relative to the lowest being one connected to gravity, ie more to the south part of the cartesian axis).

Again, walk uphill. The precise shape of stuff makes no difference.

The size of the two possible cones is regarded as equal at all times, so in effect you cannot have the same ellipse in one cone and a different sized inverted one.

You get the same ellipse from a cone as you do from an identical, inverted cone.

Of course, for reasons described, the narrator would like to believe that the highest point is indeed the one closer to the latus rectum, for reasons of practicality (less distance to travel to).

Again, there is no 'the' latus rectum. There is one through *each* focus. Whether the vertex you are closest to is the top or bottom is not defined in any way by the ellipse itself, nor by any of its properties. It's defined by which direction you have gravity pointing in.

You can do everything you want simply by trying to climb whatever the local landscape is until you reach the highest point, adding in ellipses, etc simply makes things confusing, as shown by this thread. You want to use ellipses, you should use actual interesting properties of ellipses. Like, for instance, if you have a 2D ellipse that is reflective, basically like a fence made of mirrors, and you stand at one focus and shine a torch in any horizontal direction, the beam of light will be reflected and pass through the other focus. (Something similar is how parabolic/directional microphones work).

 

[Kyriakos]

Thanks

Some clarifications:

-There is no "up and down" in the shape the narrator originally has to look at in the beginning of the story. He only observes a 2d ellipse, and that is all. There is no drawn cone at all, just the elliptical shape. In fact he even is not 100% sure (although he is mostly sure) that the ellipse has indeed been drawn in 2d in a way that shows which axis of it is the major one. (he also knows a few more minor properties, some with a relative level of near-certainty, others less so).

-I now agree fully with your point that indeed even if the two cones (inverted, upright) are identical, again one cannot tell which way is up if only left with the properties he can examine in the 2d shape. I now incorporated this "confusion" (which in reality seems to be a misguided hope) into the actual story-plotine, by making the narrator be in effect someone who locked himself away from the actual question he had to solve, into a world of an impossible problem.

-The story now is more alluding to what is going on outside the space the narrator is in (the space is a regular human room, where he observes his old notes of an ellipse, drawn on a piece of paper, with no cone drawn with it). In reality he seems to have been moving away from the actual- and more sinister, but less absurd- question he had to answer. Some other ellipse is involved, but the story is only about being shielded from the more real situation.

I am very happy with how this story ended up (nearly 10 pages) and i again want to thank all those who participated in this thread which allowed me to actually write this in a far better way- and without mistakes this time...!).
Most of all i want to thank Sanabas, cause he simply helped enormously