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 | | From: | Don H | | Subject: | Bi-conics | | Date: | Sat, 08 Jan 2005 17:46:30 GMT |
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 | The intersection of a plane with a cone gives us the "conic sections" of
plane geometry - circle, ellipse, parabola, and hyperbola. It is said: the
circle is a special case of an ellipse; but what of the other figures just
mentioned?
If, instead of a single cone, we add on underneath, its mirror-image, a
sort-of "negative" cone, with same "base" and a negative "apex"; thus
producing a composite "bi-cone", which appears diamond-shaped when viewed
side-on.
Then, both parabola and hyperbola, if extended down from their original
configuration, into this second cone, also become ellipses.
So what? Maybe this has little mathematical or other significance; but it
may be a different way of relating the four figures, bringing in negative
values in co-ordinate geometry, and a new way of determining focal points
involved.
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| | From: | Funland | | Subject: | It depends. | | Date: | Sun, 9 Jan 2005 02:08:33 +0800 |
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| Well, it can be unified the following under the 3D bi-cone.
So you are using a higher dimensional geometry in this.
Besides, the cone surface geometry can be a new branch too.
"Don H" ¦b¶l¥ó
news:W%UDd.110673$K7.34404@news-server.bigpond.net.au ¤¤¼¶¼g...
> The intersection of a plane with a cone gives us the "conic sections" of
> plane geometry - circle, ellipse, parabola, and hyperbola. It is said:
the
> circle is a special case of an ellipse; but what of the other figures just
> mentioned?
> If, instead of a single cone, we add on underneath, its mirror-image, a
> sort-of "negative" cone, with same "base" and a negative "apex"; thus
> producing a composite "bi-cone", which appears diamond-shaped when viewed
> side-on.
> Then, both parabola and hyperbola, if extended down from their original
> configuration, into this second cone, also become ellipses.
> So what? Maybe this has little mathematical or other significance; but
it
> may be a different way of relating the four figures, bringing in negative
> values in co-ordinate geometry, and a new way of determining focal points
> involved.
> ====================================
>
>
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| | From: | Cloud99992000 | | Subject: | Re: Bi-conics | | Date: | Thu, 20 Jan 2005 08:15:09 -0600 |
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| isolated from each other a negative circle is identical to a positive
circle.
how nice
Don H wrote:
>The intersection of a plane with a cone gives us the "conic sections" of
>plane geometry - circle, ellipse, parabola, and hyperbola. It is said: the
>circle is a special case of an ellipse; but what of the other figures just
>mentioned?
> If, instead of a single cone, we add on underneath, its mirror-image, a
>sort-of "negative" cone, with same "base" and a negative "apex"; thus
>producing a composite "bi-cone", which appears diamond-shaped when viewed
>side-on.
> Then, both parabola and hyperbola, if extended down from their original
>configuration, into this second cone, also become ellipses.
> So what? Maybe this has little mathematical or other significance; but it
>may be a different way of relating the four figures, bringing in negative
>values in co-ordinate geometry, and a new way of determining focal points
>involved.
>====================================
>
>
>
>
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| | From: | Don H | | Subject: | Re: Bi-conics | | Date: | Thu, 20 Jan 2005 17:58:12 GMT |
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| Isolated, yes. But I'm joining them, to form a single whole. I used terms
"negative" and "mirror-image" to help describe their juxtaposition, that's
all. The monocone produces the traditional parabola and hyperbola, but the
bicone doesn't.
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"Cloud99992000" wrote in message
news:a5PHd.43$Io.10@fe61.usenetserver.com...
> isolated from each other a negative circle is identical to a positive
> circle.
> how nice
>
> Don H wrote:
>
> >The intersection of a plane with a cone gives us the "conic sections" of
> >plane geometry - circle, ellipse, parabola, and hyperbola. It is said:
the
> >circle is a special case of an ellipse; but what of the other figures
just
> >mentioned?
> > If, instead of a single cone, we add on underneath, its mirror-image, a
> >sort-of "negative" cone, with same "base" and a negative "apex"; thus
> >producing a composite "bi-cone", which appears diamond-shaped when viewed
> >side-on.
> > Then, both parabola and hyperbola, if extended down from their original
> >configuration, into this second cone, also become ellipses.
> > So what? Maybe this has little mathematical or other significance; but
it
> >may be a different way of relating the four figures, bringing in negative
> >values in co-ordinate geometry, and a new way of determining focal points
> >involved.
> >====================================
> >
> >
> >
> >
>
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