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Re: Legitimate geometry question on cone-plane intersection

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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.

> 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?  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.  :-)

> 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.

[ .... ]

-- 
Alan Mackenzie (Nuremberg, Germany).

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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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