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Alex wrote-

| 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?

Alex is right. We have rather drifted into this discussion without
defining exactly our terms. Let me define the problem as I see it.

An observer sees a celestial body (which is at dec1, GHA1) at an
altitude of alt1, which is not near his zenith. Presumably all will
accept  that he is then somewhere on a position circle, radius (90 -
alt1), centred on the geographical position below dec1, GHA1. But we
know nothing about where he may be on that circle.

Now he travels through a known course and distance. Wherever he may be
starting from on that circle, the course and distance are the same.
The course is defined as an angle relative to true North, the normal
definition of a course that mariners use. No complication about that.
The distance is in miles, measured over the Earth's surface. The Earth
is assumed to be, for our purposes, spherical.

The question is; what is his locus then? Is it a circle? I claim not.

To avoid argument about the difference between a rhumb line course and
a great circle, you can take the simple case of  a course of true
North, which is both a rhumb line and a great circle. Even then, his
final locus is not a circle.

Is the problem now stated well enough to suit Alex? Not "the same
direction", but "the same course and distance"


| 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".

That is a tiny circle corresponding to a near-zenith observation. In
those circumstances, a small circle remains a circle when shifted.

| 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.

I disagree with that final sentence.

Consider this simple example, once again.

3 observers, A, B, and C, see a star positioned at dec = 0 degrees,
GHA = 0 degrees, at an altitude of 30 degrees. A is at 60N, 0W, B is
at 0N, 60W, C is at 45N, 45W. It is obvious that A and B are on that
circle, centred at 0N 0W, radius 60 degrees, and a bit of simple
spherical trig (or altitude tables) shows that C is also exactly on
the circle. Each observer travels due North through a distance of 60
miles. Now A is at 61N, 0W, B is at 1N, 60W, C is at 46N, 45W. Do
those 3 new positions lie on a circle? If so, where is its centre and
what is its radius? I guarantee that nobody will provide such a centre
or radius, because those positions are no longer on a circle. And the
discrepancy from a circle is not by an infinitesimal amount, either.
but by something over 14 miles, after a shift of only 60 miles.

George.

contact George Huxtable at george@huxtable.u-net.com
or at +44 1865 820222 (from UK, 01865 820222)
or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.

Geo

   

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