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Kaimbridge
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 Posted: Mon Oct 13, 2008 1:21 am Post subject: "Great Ellipse" Or "Elliptic Great Circle"? |
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The shortest distance, the "geodesic", on a sphere is along a great
circle.
On an oblate spheroid (a>b), long lines slide towards the poles as
that is the shortest path.
Take a globe and tilt it down so that the polar vertex is at the
equator level: The transverse lines of longitude define great circles
or "arc paths", "AP"s, at an even interval.
Now take a piece of string and hold one end on the equator and pull it
along the midpath (i.e., the 45° AP), all the way around to the
equator on the other side ("antipodal"). With the spherical globe,
the string will stay securely on that line all the way around.
If, however, you change the globe to an oblate spheroid and try to
pull the string from the equator along that same AP, the string will
slide off the AP towards the pole until you reach the other side of
the equator, when the string will be north-south, as that is the
shortest antipodal path.
The north-south meridian is definitively a great ellipse.
But what about the 45° AP?
Since the radius and arcradius at each point along it is different, it
is not a great circle: Is it technically a great ellipse or an
elliptic great circle... if there is a distinction between the two?
I tend to think that the path that the string physically follows (the
geodesic) is the spheroidal great ellipse, while the spherically
delineated APs are elliptic great circles.
~Kaimbridge~
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Wikipedia—Contributor Home Page:
http://en.wikipedia.org/wiki/User:Kaimbridge
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Spaceman
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| Posted: Mon Oct 13, 2008 6:38 am Post subject: Re: "Great Ellipse" Or "Elliptic Great Circle"? |
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Kaimbridge wrote:
[quote]The shortest distance, the "geodesic", on a sphere is along a great
circle.
[/quote]
Incorrect,
The shortest distance, a "straight line" is not along the geodesic at all.
I think you meant to say the shortest path along the surface.
:)
[quote]On an oblate spheroid (a>b), long lines slide towards the poles as
that is the shortest path.
Take a globe and tilt it down so that the polar vertex is at the
equator level: The transverse lines of longitude define great circles
or "arc paths", "AP"s, at an even interval.
Now take a piece of string and hold one end on the equator and pull it
along the midpath (i.e., the 45° AP), all the way around to the
equator on the other side ("antipodal"). With the spherical globe,
the string will stay securely on that line all the way around.
If, however, you change the globe to an oblate spheroid and try to
pull the string from the equator along that same AP, the string will
slide off the AP towards the pole until you reach the other side of
the equator, when the string will be north-south, as that is the
shortest antipodal path.
The north-south meridian is definitively a great ellipse.
But what about the 45° AP?
Since the radius and arcradius at each point along it is different, it
is not a great circle: Is it technically a great ellipse or an
elliptic great circle... if there is a distinction between the two?
I tend to think that the path that the string physically follows (the
geodesic) is the spheroidal great ellipse, while the spherically
delineated APs are elliptic great circles.
[/quote]
ellipse = not a perfect circle.
a great circle is a perfect circle.
mixing the two is not good for geometry.
And more important is to make sure you remember
that the shortest distance has nothing to do with either
a circle or an ellipse.
The shortest travelable path on a surface is different from
the shortest physical distance of course.
shortest path on surface of a sphere = curve
shortest physical distance on a surface on a sphere is a tunnel = straight
line.
The shortest distance from 2 degrees to 20 degrees is not
a curve at all and never will be.
:) |
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Bad Idea
Guest
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| Posted: Mon Oct 13, 2008 4:19 pm Post subject: Re: "Great Ellipse" Or "Elliptic Great Circle"? |
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There are non-mathematical definitions of "circle." Consider the term
"Great Path Along A Highly Irregular And Dynamic Surface." Perhaps
Great Circle is best after all. |
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jaf
Guest
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| Posted: Tue Oct 14, 2008 2:03 am Post subject: Re: "Great Ellipse" Or "Elliptic Great Circle"? |
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http://en.wikipedia.org/wiki/Geoid
John
"Kaimbridge" <Kaimbridge@gmail.com> wrote in message news:ff1ba022-eb93-40dd-a486-c5bec62cb415@64g2000hsu.googlegroups.com...
The shortest distance, the "geodesic", on a sphere is along a great
circle.
On an oblate spheroid (a>b), long lines slide towards the poles as
that is the shortest path.
Take a globe and tilt it down so that the polar vertex is at the
equator level: The transverse lines of longitude define great circles
or "arc paths", "AP"s, at an even interval.
Now take a piece of string and hold one end on the equator and pull it
along the midpath (i.e., the 45° AP), all the way around to the
equator on the other side ("antipodal"). With the spherical globe,
the string will stay securely on that line all the way around.
If, however, you change the globe to an oblate spheroid and try to
pull the string from the equator along that same AP, the string will
slide off the AP towards the pole until you reach the other side of
the equator, when the string will be north-south, as that is the
shortest antipodal path.
The north-south meridian is definitively a great ellipse.
But what about the 45° AP?
Since the radius and arcradius at each point along it is different, it
is not a great circle: Is it technically a great ellipse or an
elliptic great circle... if there is a distinction between the two?
I tend to think that the path that the string physically follows (the
geodesic) is the spheroidal great ellipse, while the spherically
delineated APs are elliptic great circles.
~Kaimbridge~
-----
Wikipedia—Contributor Home Page:
http://en.wikipedia.org/wiki/User:Kaimbridge
***** Void Where Permitted; Limit 0 Per Customer. ***** |
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