If a circle is merely a special case of an ellipse, then maybe an ellipse is
a special case of a circle - in which latter case, what's the justification
for an ellipse having two foci?
If the planets revolve around the sun in elliptical orbits, with the sun at
one focus (Kepler's first law), where has the other focus got to?
==============================
"John Jones" wrote in message
news:1105208946.450885.120970@c13g2000cwb.googlegroups.com...

Funland wrote:
> 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.
> > ====================================
> >
> >

Regretably I cannot help you with your geometry problem. However,
should I wear scoobydoo socks in the summer or the winter?
JJ

From:	Robin Chapman
Subject:	Re: It depends.
Date:	Mon, 10 Jan 2005 18:38:01 +0000

Don H wrote:

> If a circle is merely a special case of an ellipse,

There's no "if" about it.

> then maybe an ellipse is a special case of a circle

it isn't.

--
Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html
"Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9"
Francis Wheen, _How Mumbo-Jumbo Conquered the World_

From:	Don H
Subject:	Re: It depends.
Date:	Mon, 10 Jan 2005 19:43:21 GMT

It would seem the circle, ellipse, parabola, and hyperbola are all basically
ellipses; it is just that the parabola and hyperbola have aspects which are
outside our "normal frame of reference" ie. the mono-cone. Indeed, we could
get away from cones altogether, and generate all four geometrics from a
single ellipse, progressing from the centre (circle) along the long axis.
Parabola would come next; finally the hyperbola; tangents could be involved
in such generation.
Halley's comet orbit might be thought to be parabolic or hyperbolic, but
is really elliptical, having a period of 76 years. (Again, outside our
normal frame of reference.) A transient meteor or comet which passes the
sun once, merely has a temporary curved path; deflected from the ordinate by
gravitation.
================================
"Robin Chapman" wrote in message
news:crui03$cdl$1@newsg1.svr.pol.co.uk...
> Don H wrote:
>
> > If a circle is merely a special case of an ellipse,
>
> There's no "if" about it.
>
> > then maybe an ellipse is a special case of a circle
>
> it isn't.
>
> --
> Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html
> "Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9"
> Francis Wheen, _How Mumbo-Jumbo Conquered the World_

From:	Greg Neill
Subject:	Re: It depends.
Date:	Mon, 10 Jan 2005 21:04:53 -0500

"Don H" wrote in message
news:tVAEd.113355$K7.28493@news-server.bigpond.net.au...
> It would seem the circle, ellipse, parabola, and hyperbola are all basically
> ellipses; it is just that the parabola and hyperbola have aspects which are
> outside our "normal frame of reference" ie. the mono-cone.

No. It is algebraically obvious that they are not all the same. Compare
the equations of each. Show us how you can make

y = 1/x from a*x^2 + b*y^2 = c^2

> Indeed, we could
> get away from cones altogether, and generate all four geometrics from a
> single ellipse, progressing from the centre (circle) along the long axis.
> Parabola would come next; finally the hyperbola; tangents could be involved
> in such generation.
> Halley's comet orbit might be thought to be parabolic or hyperbolic, but
> is really elliptical, having a period of 76 years. (Again, outside our
> normal frame of reference.)

No. A repeating orbit implies an ellipse or a circle (discounting
the usual imperfections due to purturbations from other bodies).

If the orbit closes, it's not parabolic or elliptical, and the
specific energy of the body in orbit must be negative.

> A transient meteor or comet which passes the
> sun once, merely has a temporary curved path; deflected from the ordinate by
> gravitation.

From:	Don H
Subject:	Re: Definition?
Date:	Tue, 11 Jan 2005 16:52:46 GMT

With geometric shapes, as with all else, it depends on how you define the
concept involved. The mathematics follows from the definition.
Currently, it seems, circle, ellipse, parabola, and hyperbola are defined
in terms of "conic section".
Are there definitions independent of conics?
"Circle" is easy enough - but the others?
Alright, maybe we can work backwards from the algebraic to verbal
definitions?
I'd define the parabola and hyperbola, in biconic terms, as geometrics
which are outside (at least partly) the monocone. It depends somewhat, even
then, if their directional arms are envisaged as going off at a tangent, or
recombining to meet elliptically. Which is where definition is important.
======================================
"Greg Neill" wrote in message
news:huGEd.17438$TN6.615317@news20.bellglobal.com...
> "Don H" wrote in message
> news:tVAEd.113355$K7.28493@news-server.bigpond.net.au...
> > It would seem the circle, ellipse, parabola, and hyperbola are all
basically
> > ellipses; it is just that the parabola and hyperbola have aspects which
are
> > outside our "normal frame of reference" ie. the mono-cone.
>
> No. It is algebraically obvious that they are not all the same. Compare
> the equations of each. Show us how you can make
>
> y = 1/x from a*x^2 + b*y^2 = c^2
>
>
>
> > Indeed, we could
> > get away from cones altogether, and generate all four geometrics from a
> > single ellipse, progressing from the centre (circle) along the long
axis.
> > Parabola would come next; finally the hyperbola; tangents could be
involved
> > in such generation.
> > Halley's comet orbit might be thought to be parabolic or hyperbolic,
but
> > is really elliptical, having a period of 76 years. (Again, outside our
> > normal frame of reference.)
>
> No. A repeating orbit implies an ellipse or a circle (discounting
> the usual imperfections due to purturbations from other bodies).
>
> If the orbit closes, it's not parabolic or elliptical, and the
> specific energy of the body in orbit must be negative.
>
> > A transient meteor or comet which passes the
> > sun once, merely has a temporary curved path; deflected from the
ordinate by
> > gravitation.
>
>

From:	Greg Neill
Subject:	Re: Definition?
Date:	Tue, 11 Jan 2005 23:12:05 -0500

"Don H" wrote in message
news:yvTEd.114437$K7.15814@news-server.bigpond.net.au...

> I'd define the parabola and hyperbola, in biconic terms, as geometrics
> which are outside (at least partly) the monocone. It depends somewhat, even
> then, if their directional arms are envisaged as going off at a tangent, or
> recombining to meet elliptically. Which is where definition is important.

Well, you can choose to call Spain "Holland", and a
mouse "elephant", but you're going to run into
certain difficulties conversing with people who
already have working definitions of those terms.

From:	Don H
Subject:	Re: Definition?
Date:	Wed, 12 Jan 2005 15:52:42 GMT

Yes, definitions are handy things.
On my bi-conics basis, all conic sections are a type of ellipse - and I'm
sticking to that.
=================================
"Greg Neill" wrote in message
news:yr1Fd.32978$TN6.1019503@news20.bellglobal.com...
> "Don H" wrote in message
> news:yvTEd.114437$K7.15814@news-server.bigpond.net.au...
>
> > I'd define the parabola and hyperbola, in biconic terms, as geometrics
> > which are outside (at least partly) the monocone. It depends somewhat,
even
> > then, if their directional arms are envisaged as going off at a tangent,
or
> > recombining to meet elliptically. Which is where definition is
important.
>
> Well, you can choose to call Spain "Holland", and a
> mouse "elephant", but you're going to run into
> certain difficulties conversing with people who
> already have working definitions of those terms.
>
>

From:	Greg Neill
Subject:	Re: Definition?
Date:	Wed, 12 Jan 2005 23:33:23 -0500

"Don H" wrote in message
news:eJbFd.116204$K7.105028@news-server.bigpond.net.au...
> Yes, definitions are handy things.
> On my bi-conics basis, all conic sections are a type of ellipse - and I'm
> sticking to that.

Thus instantly (and very efficiently) relegating yourself
and your ideas to the realm of the ignored.

Also, your top-posting isn't helping, either.

From:	Don H
Subject:	Re: Definition?
Date:	Thu, 13 Jan 2005 15:40:32 GMT

That's alright. All new ideas take a while to percolate. I'll investigate
biconics in any case.
As to top-posting, I go along with OE which puts cursor at top LH side of a
blank line at top of any "Reply Group"; in which case previous commentaries
are footnotes; there for reference.
==================================
"Greg Neill" wrote in message
news:xRmFd.37657$TN6.1352947@news20.bellglobal.com...
> "Don H" wrote in message
> news:eJbFd.116204$K7.105028@news-server.bigpond.net.au...
> > Yes, definitions are handy things.
> > On my bi-conics basis, all conic sections are a type of ellipse - and
I'm
> > sticking to that.
>
> Thus instantly (and very efficiently) relegating yourself
> and your ideas to the realm of the ignored.
>
> Also, your top-posting isn't helping, either.
>
>