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Re: Legitimate geometry question on cone-plane intersection

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Alan Mackenzie <acm@muc.de> writes:

>Hello, Michael.

>Michael Moroney <moroney@world.std.spaamtrap.com> wrote:
>> Alan Mackenzie <acm@muc.de> writes:

>[ .... ]

>>>Construct a sphere tangent to the cone (the line of contact being a
>>>circle, of course) such that it is also tangent to the ellipse's plane.
>>>The point where it touches the ellipse is a focus.  The other focus is
>>>where the other such sphere touches the plane.

>> OK, this is the Dandelin Spheres proof that certain conic sections are an
>> ellipse.

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

>Thanks.  I didn't know this elegant construction had a name and a well
>know discoverer.

It is an elegant proof.

>> My question is whether a focus of the ellipse is on the axis of the cone.

>I think it's clear, from the above, that it's not.  Intuitively, if one
>focus was on the cone's axis, why wouldn't the other one be, too?

Yes, it is now obvious from the diagram that the cone's axis does not pass
through a focus, and cannot do so, except for the special case when the
ellipse is a circle and the cutting plane is parallel to the base.

Since the ellipse is off-center to the cone's axis, I thought that the
axis might pass through a focus instead.

>As
>one tilts the section plane from an ellipse, through a circle, back to
>an ellipse, there would be an abrupt jump from "left" to "right" in
>"which" focus was on the axis.  I agree, this would all need to be
>backed up by some maths.  :-)

A circle is centered and the center and foci are the same.

As the plane tilts the other way, the ellipse is off-center the other
side. There would be no 'jump' since the switch is at the circle when
the center and foci are the same.

>> It appears from the first diagram that neither focus is on the cone's
>> axis. My question becomes whether this point (unlabeled in the diagram
>> but it is to the right of point P) has a name or any significance, perhaps
>> in orbital mechanics.

>Intriguing question.  I don't know the answer.

So far I haven't found an answer.

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AP's thoughts as of 5JAN2019// three proofs that ellipse is never a conic Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2019-01-05 07:19 -0800
  Legitimate geometry question on cone-plane intersection moroney@world.std.spaamtrap.com (Michael Moroney) - 2019-01-05 17:48 +0000
    Re: Legitimate geometry question on cone-plane intersection Alan Mackenzie <acm@muc.de> - 2019-01-05 18:58 +0000
      Re: Legitimate geometry question on cone-plane intersection moroney@world.std.spaamtrap.com (Michael Moroney) - 2019-01-05 19:44 +0000
        Re: Legitimate geometry question on cone-plane intersection Peter Percival <peterxpercival@hotmail.com> - 2019-01-05 20:15 +0000
        Re: Legitimate geometry question on cone-plane intersection Alan Mackenzie <acm@muc.de> - 2019-01-07 03:15 +0000
          Re: Legitimate geometry question on cone-plane intersection moroney@world.std.spaamtrap.com (Michael Moroney) - 2019-01-07 04:44 +0000
      Re: Legitimate geometry question on cone-plane intersection Peter Percival <peterxpercival@hotmail.com> - 2019-01-05 20:13 +0000
        My third proof ellipse is never a conic is that the tip end of cone and bottom flared-out end can never yield a 1000to1 ellipse Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2019-01-05 13:44 -0800
          try wrapping a ellipse in a cone wrapper Re: My third proof ellipse is never a conic is that the tip end of cone and bottom flared-out end can never yield a 1000to1 ellipse Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2019-01-05 14:24 -0800
            Re: try wrapping a ellipse in a cone wrapper Re: My third proof ellipse is never a conic is that the tip end of cone and bottom flared-out end can never yield a 1000to1 ellipse Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2019-01-05 14:40 -0800
              researching if the ellipse has an internal universal constant of 1/137 Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2019-01-05 17:37 -0800
                researching if the ellipse has an internal universal constant of 1/137 Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2019-01-05 21:13 -0800
                Re: Archimedes "Village Idiot" Plutonium flunked the Math Test of a lifetime-generation test moroney@world.std.spaamtrap.com (Michael Moroney) - 2019-01-06 05:55 +0000
                not even a constant percentage in the rectangles Re: researching if the ellipse has an internal universal constant of 1/137 Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2019-01-05 21:59 -0800
                no percentage formula Re: researching if the ellipse has an internal universal constant of 1/137 Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2019-01-05 22:14 -0800
                not even rectangles Re: no percentage formula Re: researching if the ellipse has an internal universal constant of 1/137 Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2019-01-05 23:02 -0800
                directrix Re: not even rectangles Re: no percentage formula Re: researching if the ellipse has an internal universal constant of 1/137 Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2019-01-05 23:34 -0800
                understanding the Ellipse well,is the same as understanding angular momentum of physics well Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2019-01-05 23:51 -0800
                Re: understanding the Ellipse well,is the same as understanding angular momentum of physics well Me <franz.fritschee.ff@gmail.com> - 2019-01-06 03:10 -0800
          Re: My third proof ellipse is never a conic is that the tip end of cone and bottom flared-out end can never yield a 1000to1 ellipse Me <franz.fritschee.ff@gmail.com> - 2019-01-06 03:08 -0800
          Re: My third proof ellipse is never a conic is that the tip end of cone and bottom flared-out end can never yield a 1000to1 ellipse Alan Mackenzie <acm@muc.de> - 2019-01-06 11:17 +0000
            Re: My third proof ellipse is never a conic is that the tip end of cone and bottom flared-out end can never yield a 1000to1 ellipse moroney@world.std.spaamtrap.com (Michael Moroney) - 2019-01-06 18:42 +0000
  Re: AP's thoughts as of 5JAN2019// three proofs that ellipse is never a conic Jan <filmart@gmail.com> - 2019-01-06 21:47 -0800

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