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Imagine we have a sphere. And inside the sphere there are two points. Now we have an arc that is connecting these two points and it is said that arc that goes through the center of the sphere is the shortest distance between two points. For example a plane flying from point a to point b has to go little bit north/south for the shortest distance.

I asked my colleagues about this problem and all of them replied: Oh, I just solved some numerical problems and I totally understand it.

I solved a quite of them myself, but I still doesn't understand why.

I would really appreciate an explanation.

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    $\begingroup$ Arches have nothing to do with this, although arcs might. $\endgroup$
    – Olin Lathrop
    Commented Jan 7, 2018 at 14:45
  • $\begingroup$ Yep, definitely arcs ! keyboard error :) $\endgroup$
    – continuity
    Commented Jan 7, 2018 at 17:24
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    $\begingroup$ Consider you are on a latitude that is 1 mile away from the North Pole and you need to get to the opposite side of this latutude (e.g. from West to East). Because the distances are minimal, we can consider the surface of the Earth in that region approximately flat. If you go along the latitude, your total distance is 3.14 miles. If you go along the great circle throught tbe North Pole, your total diatance is only 2 miles. This example shows that latitudes are not the shortest arcs. $\endgroup$
    – safesphere
    Commented Jan 8, 2018 at 1:37

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Please refer to: https://en.wikipedia.org/wiki/Great-circle_distance

In astronomy, it should be treated as the orthodromic distance, since you are talking about planets. As the link above has pointed out, this orthodromic distance is the shortest distance between two points on the surface of a sphere, measured along the surface of the sphere (as opposed to a straight line through the sphere's interior, which you thought). It is also called the great-circle distance.

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The shortest path between two points is the straight line between them, whether these points are in a sphere or not.

Most likely there was additional context in your situation that you aren't telling us about or that you weren't aware of. It seems this context implies you want to know the shortest distance along the surface of the sphere. This is also called the great arc.

For example, if the context was airplanes flying between cities, then the straight line distance thru solid ground is not useful.

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  • $\begingroup$ Corrected my question and problem. Definitely in the context of planes going from point a to point b! Thank you for reminding me! $\endgroup$
    – continuity
    Commented Jan 7, 2018 at 17:33

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