The shortest distamce between two points

[steve_bank]

In Euclidean sapce meaning no distortions by gravity how would you derive what the shortest distance between two points is? It can not be done withinn geometry.

 

[Artemus]

Calculus of variation works:

Thus we prove that the straight line corresponds to the shortest distance
between two points.

It also works in non-Euclidean spaces.

 

[Speakpigeon]

Calculus of variation works:

Thus we prove that the straight line corresponds to the shortest distance
between two points.

It also works in non-Euclidean spaces.

At least sometimes we get to have a really good laugh around here!

Thanks Artemus!
EB

 

[phands]

It can not be done withinn geometry.

Are you saying it's not possible, or that you exclude proofs which use geometry?

Anyway.... http://www.instant-analysis.com/Principles/straightline.htm

 

[steve_bank]

It can not be done withinn geometry.

Are you saying it's not possible, or that you exclude proofs which use geometry?

Anyway.... http://www.instant-analysis.com/Principles/straightline.htm

Yes. 1st derivative min max test. I would have approached it as a line integral.

I suppose there is little point in posing problems when most can be found on the ner. I wonder if the net will lead to mental atrophy in coming generations.

 

[Jimmy Higgins]

Calculus of variation works:

Thus we prove that the straight line corresponds to the shortest distance
between two points.

It also works in non-Euclidean spaces.

Why does it have to be so "complicated"? *headache*

I understand if one goes the 3D route, it gets complicated, but isn't it easier to use another avenue of geometry, the circle? In essence, the question becomes can a circular segment be shorter than its accompanying chord?

A chord's length is approximately the square root of the circular segment length squared minus a ratio of the height of the arc squared. It is impossible for C to be less than S.

 

[Artemus]

Calculus of variation works:

Thus we prove that the straight line corresponds to the shortest distance
between two points.

It also works in non-Euclidean spaces.

Why does it have to be so "complicated"? *headache*

I understand if one goes the 3D route, it gets complicated, but isn't it easier to use another avenue of geometry, the circle? In essence, the question becomes can a circular segment be shorter than its accompanying chord?

A chord's length is approximately the square root of the circular segment length squared minus a ratio of the height of the arc squared. It is impossible for C to be less than S.

That only shows it is shorter than the circular arc paths. There are an infinite number of other paths that can followed. The only way to get the length of an arbitrary path is the line integral and the only way to minimize the length is to take the derivative and set it to zero. Thus the birth of calculus of variations.

 

[Jimmy Higgins]

Why does it have to be so "complicated"? *headache*

I understand if one goes the 3D route, it gets complicated, but isn't it easier to use another avenue of geometry, the circle? In essence, the question becomes can a circular segment be shorter than its accompanying chord?

A chord's length is approximately the square root of the circular segment length squared minus a ratio of the height of the arc squared. It is impossible for C to be less than S.

That only shows it is shorter than the circular arc paths.

You might have the math facts to back you up, but I still have my gut, and I feel pretty comfortable with my claim.

There are an infinite number of other paths that can followed. The only way to get the length of an arbitrary path is the line integral and the only way to minimize the length is to take the derivative and set it to zero. Thus the birth of calculus of variations.

Oh fine, maths for the win. Can I at least get half/infinity credit?

 

[Artemus]

You might have the math facts to back you up, but I still have my gut, and I feel pretty comfortable with my claim.

There are an infinite number of other paths that can followed. The only way to get the length of an arbitrary path is the line integral and the only way to minimize the length is to take the derivative and set it to zero. Thus the birth of calculus of variations.

Oh fine, maths for the win. Can I at least get half/infinity credit?

It was accepted axiomatically for a long, long time (at least since Euclid) so you can feel pretty safe going with your gut on this one. This is another one of the fun mathematical facts that are obvious but it took thousands of years before the techniques needed to prove it were developed. I hesitate to bring it up, but the resolution of Zeno's paradox (Achilles does pass the tortoise!) is another.

 

[steve_bank]

I had no idea the op could be so controversial.

 

[lpetrich]

In Euclidean sapce meaning no distortions by gravity how would you derive what the shortest distance between two points is? It can not be done withinn geometry.

I'll try to work out how one gets a distance value.

One starts with a space or a manifold, a set of points. It can be discrete, continuous, or some combination, like some set of continuous patches. Let us restrict ourselves to a continuous space of points, and let the points be specified with coordinates x = {x1, x2, ..., xn} or {xⁱ} (raised indices don't mean powers here but indexing).

Now consider a curve in it, some point values x(t) where t is a parameter. The curve has a distance measure along it, s(t). The distance between two points x(t1) and x(t2) is thus s(t2) - s(t1).

A "line" in such a space is a "geodesic", a curve with the shortest (or sometimes longest) distance.

Let us move two points close together. One ends up with
ds/dt = s'(t) -- some function of t.

We want to relate to the coordinates x, so we take the derivative with respect to t: w = dx/dt (wⁱ = dxⁱ/dt). That is a vector and not a scalar, and we want a scalar. So we construct a function sqrt(wⁱ*g_ij*w^j) with "metric tensor" g_ij = g_ij(x) and summing inside the square root over "dummy indices" i and j -- the "Einstein convention". This function is what we want for ds/dt.

This is constructed as a generalization of Pythagoras's theorem, and so far, g is arbitrary. g has the meaning of going from coordinate distance to true distance. One then gets geodesics by doing the "calculus of variations" on the distance integral

s = integral of sqrt(wⁱ*g_ij*w^j) over t

Our next stop is angles. For the angle a12 between tangent vectors w1 and w2 of curves 1 and 2 at some point, we generalize the familiar Euclidean-space formula:
cos(a12) = (w1ⁱ*g_ij*w2^j)/sqrt(w1ⁱ*g_ij*w1^j)/sqrt(w2ⁱ*g_ij*w2^j)

-

Let's now compare it to the axioms of Euclidean geometry. It is Neutral Geometry with Euclid's fifth postulate. Neutral geometry is a modern statement of the axioms behind Euclidean geometry, and it features these postulates:

1. The Set Postulate: lines and the space (2D: a plane) all contain points
2. The Existence Postulate: noncollinear points exist
3. The Unique Line Postulate: between every two points is a unique line
4. The Distance Postulate: every pair of points has a value for the distance between them
5. The Ruler Postulate: every line has a distance function for points along it
6. The Plane Separation Postulate: a line separates the plane into two disjoint parts
7. The Angle Measure Postulate: every pair of half-lines from a point has a value for the angle between them
8. The Protractor Postulate: every point has an angle function for half-lines coming from it
9. The SAS Postulate: for the lengths of two triangle sides and the value of the angle at their shared vertex, one can construct the rest of the triangle, and those values for that triangle are the same no matter where one constructs it.

I'll go over them, and consider how they apply to a general space.

The Set Postulate and the Existence Postulate give us a 2D space. That is not necessary in general -- one can use an arbitrary number of dimensions.

The Unique Line Postulate is true locally, bot not necessarily nonlocally.

The Distance Postulate and the Ruler Postulate are clearly satisfied.

The Plane Separation Postulate is only for 2D and it is not necessary for what we are trying to achieve -- deriving the Euclidean distance measure.

The Angle Measure Postulate is satisfied. However, the Protractor Postulate, though true in 2 dimensions, requires a coplanarity condition in more than 2 dimensions.

So far so good, or at least so it seems.

 

[lpetrich]

The final neutral-geometry postulate, the SAS postulate, has a very interesting result, but one that requires the full apparatus of Riemannian differential geometry to derive. In the limit of small triangles, it is always satisfied, but if one goes outside that limit, one finds to lowest order that this postulate is only satisfied if the curvature is "covariantly constant".

That is because, in general, the curvature is a 4-tensor, an object with 4 coordinate indices. Being covariantly constant is the closest that the curvature tensor can get to being constant while being a good differential-geometry object. It also means that the curvature has a constant overall value.


Finally, Euclid's fifth postulate. A form that easily generalizes to multiple dimensions is that a triangle's angles must add up to 180 degrees or pi radians. Calculating the sum in the fashion of checking the SAS postulate, it gives a further constraint: the curvature is zero.


There is an easy way to make a zero-curvature space: set its metric tensor g to some constant value. Diagonalizing g and absorbing scale factors into the coordinates gives g = diag(1's and -1's). One gets a Euclidean space with diag(1's) = the identity matrix. For mixed signs, one gets a pseudo-Euclidean space like flat space-time.

 

[Artemus]

I had no idea the op could be so controversial.

FWIW I thought it was quite appropriate for this subforum.