- #1

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Is a straight line the shortest possible distance from A-B or a line with no curves?

Bit of an annoying question, I know.

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- #1

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Is a straight line the shortest possible distance from A-B or a line with no curves?

Bit of an annoying question, I know.

- #2

fresh_42

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A straight line is usually what your friend said: no curvature, Euclidean space of any dimension. What is wrong, is the phrase: "A straight line is the shortest distance between two points." Actually the shortest distance between two points is a geodesic, which happens to be a straight line in Euclidean geometries and a curvature in non Euclidean geometries like earth. One can easily see it when we look a flight tracking websites. But a line is a one dimensional affine space in a Euclidean vector space.

If you should call geodescis a straight line, which you might do as a convention, then you have to make it clear. It is not the usual convention.

- #3

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It turns out that in most cases you are likely to encounter, that definition of 'no curves' also minimises the distance travelled (subject to the applicable constraints, which in this case means staying on the surface of the sphere), so that it is the same as defining it as the path with the shortest distance.

- #4

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I would not want to be onboard a plane taking a 'straight line' from London to Tokyo

- #5

fresh_42

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Why not? I don't see a problem, except they are not really fast ...I would not want to be onboard a plane taking a 'straight line' from London to Tokyo

(http://www.viatoura.de/images/fotoalbum/fotoalbum6/05_koelner-bucht_region.jpg)

- #6

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Yeah, and the flight attendants are ugly.Why not? I don't see a problem, except they are not really fast ...

- #7

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A straight line is usually what your friend said: no curvature, Euclidean space of any dimension. What is wrong, is the phrase: "A straight line is the shortest distance between two points." Actually the shortest distance between two points is a geodesic, which happens to be a straight line in Euclidean geometries and a curvature in non Euclidean geometries like earth. One can easily see it when we look a flight tracking websites. But a line is a one dimensional affine space in a Euclidean vector space.

If you should call geodescis a straight line, which you might do as a convention, then you have to make it clear. It is not the usual convention.

If objects travelling in a constant speed on a geodesic have zero proper acceleration, then surely it must mean that geodesics are straight lines.

- #8

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If objects travelling in a constant speed on a geodesic have zero proper acceleration, then surely it must mean that geodesics are straight lines.

A "straight line" is what a straight line is defined to be. You're falling into the trap of expecting mathematical objects to have properties that are suggested by what you call them.

For example, in the past some very able mathematicians spent their lives trying to prove Euclid's 5th (parallel) postulate from the first four. It never occurred to them that the first four postulates needn't specify what they considered to be a "straight line". So, they tied themselves in knots confusing mathematics with everyday language and concepts.

In the end, it was recognised that the first four axioms fitted a wider class of objects - including lines on the surface of a sphere. Whether you consider lines on the surface of a sphere to be "straight" or not becomes, IMHO, a matter of definition. It's the same with geodesics in general. They are straight lines if you define "straight" to mean "geodesic" and they are not straight lines if you reserve that term for Euclidean geometry.

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