What is the definition of a straight line in mathematics?

[Lunct]

Really basic question. I was talking with my friend and we started to get onto discussing lines when I said in three dimensions a straight line can be curved. He thinks of a straight line as a line with no curve, whilst I see it as the shortest possible distance from A to B, which in 3 dimensions can be curved. I think that he is talking about 2-dimensional Euclidean geometry, and is not factoring in a 3rd dimension.

Is a straight line the shortest possible distance from A-B or a line with no curves?Bit of an annoying question, I know.

 

[fresh_42]

First of all, this has nothing to do with dimensions, except we need at least dimension one to speak about lines.

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.

 

[andrewkirk]

It is usually defined as 'having no curves' (writing that formally uses some pretty fancy mathematics and is a bit intimidating), but that doesn't mean what it sounds like. For instance, if the path is constrained to the surface of a sphere, 'no curves' means no deviations to left or right, and that gives a path called a 'great circle', which is the ideal path for an aeroplane traveling between two cities. Viewed from outside the sphere, the great circle path looks curved, because the sphere is.

It turns out that in most cases you are likely to encounter, that definition of 'no curves' also minimises the distance traveled (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.

 

[rootone]

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

 

[fresh_42]

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

Why not? I don't see a problem, except they are not really fast ...

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

 

[rootone]

Why not? I don't see a problem, except they are not really fast ...

Yeah, and the flight attendants are ugly.

 

[Lunct]

First of all, this has nothing to do with dimensions, except we need at least dimension one to speak about lines.

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

 

[PeroK]

If objects traveling 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.