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01-01-2020
11:09 AM

01-01-2020
11:09 AM

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1 ACCEPTED SOLUTION

Accepted Solutions

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01-02-2020
09:53 AM

01-02-2020
09:53 AM

Yes,

here is example when a=15, b=8 and alpha=30 degree...

..and when alpha changes to 40 degree

Observe that auxiliary view is normal to cutting plane and does not show real cone angle.

11 REPLIES 11

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01-02-2020
08:16 AM

01-02-2020
08:16 AM

Re: Cone, plane, ellipse

a & b are not sufficient to define alpha.

No mather what alpha you have, you can allways slice cone to obtain ellipse with dimmensions a & b.

/Avdo

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01-02-2020
09:22 AM

01-02-2020
09:22 AM

Re: Cone, plane, ellipse

Thanks!

For a of a **circular** cone too?

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01-02-2020
09:53 AM

01-02-2020
09:53 AM

Yes,

here is example when a=15, b=8 and alpha=30 degree...

..and when alpha changes to 40 degree

Observe that auxiliary view is normal to cutting plane and does not show real cone angle.

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01-02-2020
10:32 AM

01-02-2020
10:32 AM

Re: Cone, plane, ellipse

Thanks!

I suspected it. But I always want to check!

When I was in school, I drew something around 1000 triangles, measured their angles and made sure that their sum is 200 degrees.

Just kidding! 180 degrees!

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01-02-2020
03:27 PM

01-02-2020
03:27 PM

Re: Cone, plane, ellipse

Nice,

Now: is it possible to prove, or dis-prove, that one of the two focal points of the ellipse lies on the axis of the cone?

Luc

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01-02-2020
04:00 PM

01-02-2020
04:00 PM

Re: Cone, plane, ellipse

Fine idea!

Add condition of the problem!

One of the two focal points of the ellipse lies on the axis of the cone.

PS

New year tree for our problem

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01-02-2020
07:14 PM

01-02-2020
07:14 PM

Re: Cone, plane, ellipse

@ValeryOchkov wrote:

Fine idea!

Add condition of the problem!

One of the two focal points of the ellipse lies on the axis of the cone.

Mission impossible (if we still are talking about a circular cone).

The focal points are where the Dandelin spheres touch the plane.

-> https://en.wikipedia.org/wiki/Dandelin_spheres

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01-03-2020
01:57 AM

01-03-2020
01:57 AM

Re: Cone, plane, ellipse

Theoretically there is one trivial solution to the problem!

One focus is on the axis if other focus is on axis too 🙂 lol

observe that circle is only special variant of the ellipse when a=b=r

otherwise it is allways offset from cone axis to the focal points.

/Avdo

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01-03-2020
09:18 AM

01-03-2020
09:18 AM

Re: Cone, plane, ellipse

@ptc-5737066 wrote:

...

otherwise it is allways offset from cone axis to the focal points.

Which needs to be proved.

One way to do so is by the Dandelin spheres - if of them should touch the plane in a point on the axis, the plane has to be horizontal and the intersection is a circle.

You sure know that the center of the ellipse is not on the axis as well (unless in the case of a circle).

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01-03-2020
10:02 AM

01-03-2020
10:02 AM

Re: Cone, plane, ellipse

Yes!

When Dandelin spheres touch eachother both focuses and ellipse center melt into touch point onto axe and ellipse becommes circle.

Since that no focus points nor ellipse center can coincident with axis.

Axis is allways intersecting the line between two focuses splitting it in one longer and one shorter part.

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01-04-2020
06:04 AM

01-04-2020
06:04 AM

Re: Cone, plane, ellipse

Hi,

With the use of analytical geometry and with the data available, this is all you can do:

Happy New Year.

F. M.