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| From | moroney@world.std.spaamtrap.com (Michael Moroney) |
|---|---|
| Newsgroups | sci.math |
| Subject | Re: Legitimate geometry question on cone-plane intersection |
| Date | 2019-01-05 19:44 +0000 |
| Organization | The World : www.TheWorld.com : Since 1989 |
| Message-ID | <q0r1et$v1k$1@pcls7.std.com> (permalink) |
| References | <17674064-528b-436c-abef-e44ddb32429d@googlegroups.com> <q0qqlf$94n$1@pcls7.std.com> <q0qup1$1ncu$1@news.muc.de> |
Alan Mackenzie <acm@muc.de> writes:
>Hello, Michael.
>Michael Moroney <moroney@world.std.spaamtrap.com> wrote:
>> I have a geometry question for anyone who knows geometry regarding the
>> cone-plane intersection ellipse. Given:
>> ?
>> ? ? ? ?/ ?\A
>> ? ?Y/ ?C ?\X
>> ? B/ ? ? ? ? \
>> ?
>> Since the angle AXC must be less than 90 degrees and the angle BYC must be
>> greater than 90 degrees and the length XC = YC, segment AC must be less
>> than BC. Therefore the center of the ellipse cannot correspond to the axis
>> of the cone, but AB is the major axis of the ellipse with point C lying on
>> the major axis.
>> Does point C correspond to one of the foci of the ellipse? If so, does
>> anyone know the name of a mathematical proof of this? If not, does point C
>> have a name, and what is the formula for finding the distance from the
>> center of the ellipse to the point C?
>I'm sorry that my newsreader can't display your diagram clearly enough
>for me to make sense of it, but I know the answer to your question -
>this is from a (British) Open University maths television program from,
>I suppose, the mid to late 1970s.
How about this:
^
/ \A
Y/ C \X
B/ \
/ \
>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
My question is whether a focus of the ellipse is on the axis of the cone.
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.
>Extend the plane through the circle of contact of the sphere and cone
>till it intersects with the ellipse's plane. The line of intersection
>is the directrix.
>I can't remember the exact details of the proof, but it was fairly
>simple, depending only on getting the appropriate projections to do the
>reasoning.
https://en.wikipedia.org/wiki/Dandelin_spheres explains it quite well,
other than glossing over why the distance P-F1 is equal to the distance
P-P1.
>The above also works for a hyperbola, parabola, and circle (sort of - it
>has no directrix).
Interesting that this proof works for other conic sections as well.
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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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