I am having a hard time grasping this concept. I understand that between two points, the great circle is the shortest point on a sphere, but why? In this picture, for example, wouldn't the shortest distance be a symmetrical line instead of a curve that the plane follows.
http://en.wikipedia.org/wiki/File:Great-…
http://en.wikipedia.org/wiki/File:Great-…
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You are confusing 2 dimensions with 3 dimensions.
try to imagine wrapping the flat paper map of the world onto a ball.
If you look in any good quality atlas you may see that in order for a world map to wrap onto a sphere you need to cut segments away from the top and the bottom at the poles.
Cartographers (map makers) give us the flat image using what is called a mercator projection. If you look closely at big scale maps of the oceans you will see the spacing at the sides increases the closer you go to the poles. It therefore distorts the distances we see visually, but makes it easier for navigators to comprehend. If you were to look at a chart of the polar regions (a gnomic chart) it looks like a wheel with all the spokes in.
The best way to understand is to try to do this. Find a ball.Wrap it in paper around the centre (equator) and cut the paper ( so the width of the paper it in effect the circumference of the ball) Draw a straight line on the paper.Now cut v shapes out of the top and bottom so it wraps closely on the top and bottom. Is your straight line still straight?? No, of course it is curved.This is the origin of the great circle concept. navigators used advanced mathematics, extensions of sine, cosine and tangents, and call it spherical trigonometry, basically doing the same calculations with curved lines.
try to imagine wrapping the flat paper map of the world onto a ball.
If you look in any good quality atlas you may see that in order for a world map to wrap onto a sphere you need to cut segments away from the top and the bottom at the poles.
Cartographers (map makers) give us the flat image using what is called a mercator projection. If you look closely at big scale maps of the oceans you will see the spacing at the sides increases the closer you go to the poles. It therefore distorts the distances we see visually, but makes it easier for navigators to comprehend. If you were to look at a chart of the polar regions (a gnomic chart) it looks like a wheel with all the spokes in.
The best way to understand is to try to do this. Find a ball.Wrap it in paper around the centre (equator) and cut the paper ( so the width of the paper it in effect the circumference of the ball) Draw a straight line on the paper.Now cut v shapes out of the top and bottom so it wraps closely on the top and bottom. Is your straight line still straight?? No, of course it is curved.This is the origin of the great circle concept. navigators used advanced mathematics, extensions of sine, cosine and tangents, and call it spherical trigonometry, basically doing the same calculations with curved lines.
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It is not the shortest point. A point has no length. The great circle is an arc along the surface of a sphere. It has length. And it is the shortest distance between two points, if restricted to the surface.
A symmetrical line? If you will notice, the plane has to go around Russian and Japanese airspace during its flight. That is why the line is not uniformly curved on that map. And the curve the jet airplane follows is probably a great circle segment. It is the flattening of the globe onto a sheet that distorts the great circle.
A symmetrical line? If you will notice, the plane has to go around Russian and Japanese airspace during its flight. That is why the line is not uniformly curved on that map. And the curve the jet airplane follows is probably a great circle segment. It is the flattening of the globe onto a sheet that distorts the great circle.
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A great circle IS symetrical...
The real problem is that the map you are looking at is a mercator projection (it preserves direction, not distance, which makes it very useful for most purposes). On it, great circles tend to be straight lines (shortest distance between points on a plane), rather than curves, so, in a way, you are right. If you measured the distance on a globe, you would find that the shortest distance between two point is always on a great circle (preserving both direction and distance)..
The real problem is that the map you are looking at is a mercator projection (it preserves direction, not distance, which makes it very useful for most purposes). On it, great circles tend to be straight lines (shortest distance between points on a plane), rather than curves, so, in a way, you are right. If you measured the distance on a globe, you would find that the shortest distance between two point is always on a great circle (preserving both direction and distance)..