Kepler's First Law

"Each planet moves in an orbit that is an ellipse with the Sun at one
focus."
Presumably this is an approximation, which is acceptable for all practical
purposes. Where two (or more) masses are linked, like binary stars, they
tend to rotate around a common barycentre - unless, like the Sun, the mass
of one predominates. Earth-and-Moon (81:1) provides a better example of
barycentre rotation. The Sun-and-planets combination is complicated by
having many bodies involved. Sun-and-Earth (333,000:1)
====================================

"Don H" wrote in message
...
"Each planet moves in an orbit that is an ellipse with the Sun at one
focus."
Presumably this is an approximation, which is acceptable for all practical
purposes.

Unless one is looking for very high precision or at extreme
situations. The precession of the perihelion of Mercury
comes to mind.

Where two (or more) masses are linked, like binary stars, they
tend to rotate around a common barycentre - unless, like the Sun, the mass
of one predominates.

There is a common barycentre for any set of masses, whether or
not one mass predominates.

Earth-and-Moon (81:1) provides a better example of
barycentre rotation. The Sun-and-planets combination is complicated by
having many bodies involved. Sun-and-Earth (333,000:1)

Don H wrote:
Thanks. Presumably same applies at sub-atomic level between nucleus
of
proton/neutron and circling electons.
Also, consider - "Bi-conics":
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.
==================================


http://mathworld.wolfram.com/ConicSection.html


Double-A

Thanks for website reference. The inverted cone as illustrated on that
site, I've merely taken and added to base of the lower cone - then both
parabola and hyperbola are converted into ellipses. They do not thereby
lose their uniqueness if we consider the mono-cone as our normal "frame of
reference". But it does demonstrate the commonality of all four conic
sections; all being a type of ellipse.
====================================
"Double-A" wrote in message
ups.com...

Don H wrote:
Thanks. Presumably same applies at sub-atomic level between nucleus
of
proton/neutron and circling electons.
Also, consider - "Bi-conics":
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.
==================================


http://mathworld.wolfram.com/ConicSection.html


Double-A

Don H wrote:
Thanks for website reference. The inverted cone as illustrated on that
site, I've merely taken and added to base of the lower cone - then both
parabola and hyperbola are converted into ellipses. They do not thereby
lose their uniqueness if we consider the mono-cone as our normal "frame of
reference". But it does demonstrate the commonality of all four conic
sections; all being a type of ellipse.
====================================
"Double-A" wrote in message
ups.com...

Don H wrote:

Thanks. Presumably same applies at sub-atomic level between nucleus

of

proton/neutron and circling electons.
Also, consider - "Bi-conics":
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.
==================================


http://mathworld.wolfram.com/ConicSection.html


Double-A




You had better check the join of the two conic sections. Neither will
form an elliptical shape as neither is normal to the semi axis at any
point.

Dave N.

"Don H" wrote in message
...
Thanks for website reference. The inverted cone as illustrated on that
site, I've merely taken and added to base of the lower cone - then both
parabola and hyperbola are converted into ellipses.

Note: tacking together the open ends of two partial parabola/hyperbola does
not an ellipse make.


They do not thereby
lose their uniqueness if we consider the mono-cone as our normal "frame of
reference". But it does demonstrate the commonality of all four conic
sections; all being a type of ellipse.
====================================

"You had better check the join of the two conic sections. Neither will form
an elliptical shape as neither is normal to the semi axis at any point."
Maybe. But the curve of parabola or hyperbola is confined to the bicone,
and cannot extend beyond it; hence parabola and hyperbola become ellipses
under this limitation. Also, the inferior cone is a "mirror image" of the
superior.
Of course, if it is envisaged that the directional arms of parabola and
hyperbola (under monocone conditions) go off tangentially, rather than
recombining, then OK. It's a matter of definition.
At present, the four (circle, ellipse, parabola, hyperbola) tend to be
defined in term of "conic section". Is an independent definition possible?
"Circle" is easy enough, but the others?
=================================
"David G. Nagel" wrote in message
...
Don H wrote:
Thanks for website reference. The inverted cone as illustrated on that
site, I've merely taken and added to base of the lower cone - then both
parabola and hyperbola are converted into ellipses. They do not thereby
lose their uniqueness if we consider the mono-cone as our normal "frame
of
reference". But it does demonstrate the commonality of all four conic
sections; all being a type of ellipse.
====================================
"Double-A" wrote in message
ups.com...

Don H wrote:

Thanks. Presumably same applies at sub-atomic level between nucleus

of

proton/neutron and circling electons.
Also, consider - "Bi-conics":
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.
==================================


http://mathworld.wolfram.com/ConicSection.html


Double-A




You had better check the join of the two conic sections. Neither will
form an elliptical shape as neither is normal to the semi axis at any
point.

Dave N.

"Note: tacking together the open ends of two partial parabola/hyperbola does
not an ellipse make."
Alright, but it does make a shape, and maybe needs defining?
==================================
"John Zinni" wrote in message
. ..
"Don H" wrote in message
...
Thanks for website reference. The inverted cone as illustrated on that
site, I've merely taken and added to base of the lower cone - then both
parabola and hyperbola are converted into ellipses.

Note: tacking together the open ends of two partial parabola/hyperbola
does
not an ellipse make.


They do not thereby
lose their uniqueness if we consider the mono-cone as our normal "frame
of
reference". But it does demonstrate the commonality of all four conic
sections; all being a type of ellipse.
====================================

"Don H" wrote in message
...
"Note: tacking together the open ends of two partial parabola/hyperbola
does
not an ellipse make."
Alright, but it does make a shape, and maybe needs defining?

It's already very well defined ... the open ends of two partial
parabola/hyperbola tacking together.

What is it that needs defining???

"Don H" wrote in message
...
"You had better check the join of the two conic sections. Neither will
form
an elliptical shape as neither is normal to the semi axis at any point."
Maybe. But the curve of parabola or hyperbola is confined to the
bicone,
and cannot extend beyond it; hence parabola and hyperbola become ellipses
under this limitation.

They most certainly do not!!!


Also, the inferior cone is a "mirror image" of the
superior.
Of course, if it is envisaged that the directional arms of parabola and
hyperbola (under monocone conditions) go off tangentially, rather than
recombining, then OK. It's a matter of definition.
At present, the four (circle, ellipse, parabola, hyperbola) tend to be
defined in term of "conic section". Is an independent definition
possible?
"Circle" is easy enough, but the others?
=================================