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Dear George,

You seem to discuss an interesting mathematical problem
but I am not sure that I could extract the exact statement
of the problem from your message.

Apparently you ask this:
Suppose we have a circle on the surface of the sphere.
Then we move each point of this circle by a fixed distance
in the SAME GIVEN DIRECTION.
Will the resulting figure be a circle?

1. This problem is not well stated until you specify exactly
what a "given direction" means.
In navigation, we usually understand by "direction" the angle
from the meridian passing through the point where we stand.

Say, N direction or NW direction.
But troubles happen near the poles.

At a pole itself, this notion of direction has no sense at all.
Near the pole, it has a sense, but this sense is different
from what we usually mean by moving in a given direction.

Suppose you are 1 mile from the N pole, longitude 0.
And you move 10 yards  West. Another guy starts
at longitude 180, also 1 mile from the pole, and also
moves 10 yards West. Do you think you and this guy move
"in the same direction"?

If this is what you really mean by "the same direction",
the shifted circle will NOT be a circle anymore.
And the size of the circle is irrelevant here.
(Relevant is how far it is from a pole).

Consider what will happen to a circle of radius 1 mile,
centered 1 mile from the pole, if you move each point of
the circle 1 mile to the West (that is 1 mile along the
parallel passing through this point).
The circle will become terribly distorted, the part near the pole
will become a spiral winding infinitely many times around the pole.

There are a few exceptional circles which will remain circles,
but they are really exceptional.
For example take a parallel.
If you shift every point of it some distance N or S (or even NW)
you will obtain another parallel, which is a circle.

2. Now we can ask the question: can one define rigorously,
without using meridians, or other arbitrary lines substituting them,
what does it mean that "every point of some circle is shifted in the SAME
DIRECTION? The answer is NO. "Same direction" cannot be defined
consistently for different points on the sphere. (This was essentially
discovered by Gauss). Whatever system of lines you use
to measure the direction, these lines will meet at some
points and cause the troubles described above.

3. Now back to Navigation. If you are well away from the poles,
a piece of the line of position near you is also away
from the poles. If you shift every point of this piece
of the position line by some moderate distance
(=MUCH SMALLER that the distance from the poles),
in the "same direction", as measured with respect to meridians,
the result will be APPROXIMATELY a piece of a circle.
Almost never will it be an exact arc of a circle.

4. Usually, this piece of a circle can be safely replaced
by a piece of a straight line on a Mercator map.
(As we almost always do in practice).

5. Now, when exactly is the use of a circle of position
justified? When the radius of this circle is small.
(That is you observe a body close to zenith).
If you are far from the pole, you can safely shift
each point of this small circle, say 10 miles NW
and the resulting figure will be a circle very nearly.
But not exactly.

6. The difference between "very nearly" and "exactly"
makes sense only in pure mathematics.
We know that the Earth surface is not exactly a sphere,
and so on. The only practical question to ask about these
idealizations is the magnitude of error we make
using an idealization.

If your circle of position has radius 60 miles,
and you are at latitude 40,
you can shift each point of your circle 10 miles W
and assume that the result will be a circle
"for all practical purposes".

7. Conclusion. To the mathematical
problem stated in the beginning the answer is "no".
The shifted line of position will not be a circle.
In practice,
if the original circle is far enough from the poles
the shifted line of position will be a circle very nearly.

Alex.

   

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