Given an ellipse, determine how a cone was sliced to create it

[Tsunamibomb]

An ellipse is a conic section, i.e. it can be described as the surface of a cut through a cone. Given an ellipse, I want to calulate the properties of its corresponding cone and how it was cut out of it. I might later follow up with more questions about the relationship of an ellipse to a cone, but here we start:

- The end points of the major axis of the ellipse are points on the cone's surface (mantle) at different heights. What is the (circular) diameter of the cone at each of those two points?

- Given the proportions of the semi and major axes of the ellipse, what can be said about the apex angle of the cone, and the angle of the elliptic conic section plane to the cone's circular base plane?

Intuitively, I suppose that a combination of the cone's apex angle and the angle of the conic section cut to the base plane of the cone, might yield the same ellipse. If so, I'd like to know how the combination of those two properties of the cone can be restricted, given the properties of an ellipse.

 

[Cubist]

I've added some labels to your image that might help with this discussion...


I assume that the "-" points in your text are copied from a question. If so, then please can you provide the full original question?

- The end points of the major axis of the ellipse are points on the cone's surface (mantle) at different heights. What is the (circular) diameter of the cone at each of those two points?

A sketch of the cross section that contains points A,B,F,C and G might be useful to answer this. Could you provide a sketch?

- Given the proportions of the semi and major axes of the ellipse, what can be said about the apex angle of the cone, and the angle of the elliptic conic section plane to the cone's circular base plane?

Can you provide another cross sectional sketch that would be useful to help answer this?

 

[Tsunamibomb]

I've added some labels to your image that might help with this discussion...

View attachment 35288
I assume that the "-" points in your text are copied from a question. If so, then please can you provide the full original question?


A sketch of the cross section that contains points A,B,F,C and G might be useful to answer this. Could you provide a sketch?


Can you provide another cross sectional sketch that would be useful to help answer this?

This is just for my own understanding of the relationship between an ellipse and a cone. I myself don't see much of tidy maths describing their relationsship, and I can't find much about it either. AFAIK there isn't even an analytical formula for the circumface of an ellipse, so these creatures are weirder than they look.

The end points of the major axis would be points B and C according to your labels, if there was any doubts about that. The "-" are meant to be bullet points for my specific questions. It is about generic ellipses and cones, so there's not much more to be sketched about it.

 

[NotJeffM]

This is just for my own understanding of the relationship between an ellipse and a cone. I myself don't see much of tidy maths describing their relationsship, and I can't find much about it either. AFAIK there isn't even an analytical formula for the circumface of an ellipse, so these creatures are weirder than they look.

The end points of the major axis would be points B and C according to your labels, if there was any doubts about that. The "-" are meant to be bullet points for my specific questions. It is about generic ellipses and cones, so there's not much more to be sketched about it.

I want to point out that the conic sections sre defined relative to a right cone.

 

[Dr.Peterson]

An ellipse is a conic section, i.e. it can be described as the surface of a cut through a cone. Given an ellipse, I want to calculate the properties of its corresponding cone and how it was cut out of it. I might later follow up with more questions about the relationship of an ellipse to a cone, but here we start:

...

Intuitively, I suppose that a combination of the cone's apex angle and the angle of the conic section cut to the base plane of the cone, might yield the same ellipse. If so, I'd like to know how the combination of those two properties of the cone can be restricted, given the properties of an ellipse.
View attachment 35286

The cone you are looking for is not unique; there will be a family of cones. (This is obvious in at least the special case of the circle, where any point on the normal line through the center of the circle will be the vertex of a cone generating this circle.) Is that what you were suggesting in the last line?

I believe you will find an interesting result if you find the locus of the vertex of these cones. (I think it will be a conic section, probably a hyperbola.)

One way to generate the family of such cones in general is the concept of Dandelin spheres:

These are tangent to the cone, and to the plane of the conic section at its foci.

Imagine a fixed ellipse as shown, and suppose you set a sphere on one focus F₁, with variable radius F₁M. You should be able to locate the vertex of the cone, S, using the vertices of the ellipse; and as you change the radius of the sphere, you will find a family of different cones, all of which intersect the plane in the same ellipse.

 

[Cubist]

It is about generic ellipses and cones, so there's not much more to be sketched about it.

In my experience a well labelled clear sketch is always a good starting point for geometry problems. However, if you wish to find answers in a different way (using words only?) then I personally won't be giving you any further help.

 

[Dr.Peterson]

An ellipse is a conic section, i.e. it can be described as the surface of a cut through a cone. Given an ellipse, I want to calculate the properties of its corresponding cone and how it was cut out of it. I might later follow up with more questions about the relationship of an ellipse to a cone, but here we start: ...

- Given the proportions of the semi and major axes of the ellipse, what can be said about the apex angle of the cone, and the angle of the elliptic conic section plane to the cone's circular base plane?

Intuitively, I suppose that a combination of the cone's apex angle and the angle of the conic section cut to the base plane of the cone, might yield the same ellipse. If so, I'd like to know how the combination of those two properties of the cone can be restricted, given the properties of an ellipse.

In my spare time I worked on proving my conjecture, which turned out to be correct: the apex of the cone can lie anywhere on a hyperbola whose vertices are the foci of the ellipse, and whose foci are the vertices of the ellipse. So if the ellipse, in the xy-plane, is x^2/a^2 + y^2/b^2 = 1, then the apex lies on x^2/c^2 - z^2/b^2 = 1, in the xz-plane, where c^2 = a^2 - b^2.

From this, you could derive information about the relationship of the angles in the cone; I didn't dig deeply into this, but if I didn't make a mistake, the tangents of the apex angle of the cone (from the axis to a generator) and the angle of the cutting plane (relative to the axis) have the ratio c/a. Intriguing. I don't think I've ever seen such information.

 

[blamocur]

In my spare time I worked on proving my conjecture, which turned out to be correct: the apex of the cone can lie anywhere on a hyperbola whose vertices are the foci of the ellipse, and whose foci are the vertices of the ellipse. So if the ellipse, in the xy-plane, is x^2/a^2 + y^2/b^2 = 1, then the apex lies on x^2/c^2 - z^2/b^2 = 1, in the xz-plane, where c^2 = a^2 - b^2.

From this, you could derive information about the relationship of the angles in the cone; I didn't dig deeply into this, but if I didn't make a mistake, the tangents of the apex angle of the cone (from the axis to a generator) and the angle of the cutting plane (relative to the axis) have the ratio c/a. Intriguing. I don't think I've ever seen such information.

Pretty cool! I haven't seen this before either. I wonder if this relation between an ellipse and a hyperbola is symmetric. I.e., given a hyperbola what is the loci of the corresponding cones? And how would this work for parabolas?

 

[Dr.Peterson]

Pretty cool! I haven't seen this before either. I wonder if this relation between an ellipse and a hyperbola is symmetric. I.e., given a hyperbola what is the loci of the corresponding cones? And how would this work for parabolas?

I think it is, indeed, more general. I know it works for parabolas, because the inspiration for my conjecture was an animation of the parabola that was provided in a comment on my blog, which looks like this:

​

The red parabola is the locus of the vertex of the cone generating the green parabola. Again, we have the vertex and focus interchanged. I didn't prove that fact at the time; and I would be interested in a more geometric proof!