View Full Version : What is the shortest distance between two points?

Vaelroth

2006-Mar-22, 01:37 AM

Seriously now, what is the shortest distance between two points?

Wrong Answers:

Slam-Dunk

Zero

Answers that aren't quite wrong, but aren't right:

Straight Line

Geodesic

Go!

Fr. Wayne

2006-Mar-22, 09:54 AM

1 point.

worzel

2006-Mar-22, 01:16 PM

Are we talking about points in spacetime? Isn't it then dependant on the frame of reference and therefore arbitrarily small?

farmerjumperdon

2006-Mar-22, 02:00 PM

In Plane Geometry, it is a straight line.

In what kind of geometry is it not a straight line? This is not a challenge, but a genuine question. (I never went past Plane Geometry).

Gruesome

2006-Mar-22, 02:18 PM

In Plane Geometry, it is a straight line.

In what kind of geometry is it not a straight line? This is not a challenge, but a genuine question. (I never went past Plane Geometry).

Anyone would have a hard time convincing me it's something other than a straight line in our normal, everyday three dimensional construct.

It might be something else in some wierd, obscure geometric theory, but I'd have a tough time buying it.

epenguin

2006-Mar-22, 02:37 PM

Don't mess around, just go straight there without stopping at any other points on the way, don't even go through any points on the way if you can help it.

WaxRubiks

2006-Mar-22, 02:37 PM

well the shortest distance between two points on a sphere is a curved line in 3d but I suppose it is just a straight line in the geometery of the surface of that sphere.

farmerjumperdon

2006-Mar-22, 02:41 PM

Exactly my feeling - which is all it is because I don't know.

But if someone came up with the answer that in curved space it would be a curved line - I'd be OK with that. In a way though, the straight line answer still stands - just with a little addendum:

The shortest distance between any 2 points is the straightest line possible, given that the line might need to curve based on the curveture of space, but the curve would never be more than the curvature of space.

In essence, it still appears to be a straight line to me. It probably appears to be a straight to an observer too.

farmerjumperdon

2006-Mar-22, 02:43 PM

well the shortest distance between two points on a sphere is a curved line in 3d but I suppose it is just a straight line in the geometery of the surface of that sphere.

I don't think so. I think the shortest distance between 2 points on a sphere is a straight line that has part of it's length inside the sphere.

The line intersects the sphere at only the 2 given points.

Gruesome

2006-Mar-22, 03:05 PM

well the shortest distance between two points on a sphere is a curved line in 3d but I suppose it is just a straight line in the geometery of the surface of that sphere.

I agree.

Consider the attached image. The shortest distance between points A and B is the straight line. However, if A is Chicago and B is London, one can't exactly fly a plane through the Earth. But the straight line is still shortest.

farmerjumperdon

2006-Mar-22, 03:15 PM

Right, nobody said it had to be the easiest line to travel.

We recommend all passengers remain in their seats as we will be plunging thru the Earth for the duration of our flight.

WaxRubiks

2006-Mar-22, 03:24 PM

Yes but using the surface geometry of the sphere, the shortest distance is a straight line along the surface that is a curved line in 3Dspace.

farmerjumperdon

2006-Mar-22, 03:38 PM

Yes but using the surface geometry of the sphere, the shortest distance is a straight line along the surface that is a curved line in 3Dspace.

But that is putting in qualifiers that change the whole question. Kinda like:

What is the shortest distance between 2 points on opposite sides of a cube?

The answer is a stright line that goes thru the cube and intersects it at the 2 named points. Taking into account the geometry of the cube doesn't change the question or the answer - unless you specify the answer is limited to those solutions constructed of line segments that never leave the planes out of which the cube is constructed.

In that case you are prescribing a solution so specialized that it has lost meaning to the original question - which appears to be a search for some sort of fundamental principle.

WaxRubiks

2006-Mar-22, 03:55 PM

well what about a blackhole?

if that streaches space-time, what is the shortest distance between two points in space on opposite sides of the blackhole?

Moose

2006-Mar-22, 04:07 PM

In computer science, the question's a little more complex, because the environment isn't necessarily going to conform to what we can expect of our run-of-the-mill three dimensions.

For example, we know that the straight real-world path between points A and B will necessarily be shorter (or equal) to one that goes first to C and then to B.

But in CS, A->B may well end up being "longer" than A->C->B or even B->A for whatever definition of "longer" you're considering. And your segments won't necessarily be equivalently bi-directional. A->B might be 5 units long while B->A might be 20 units long. Or only 1 unit long. Or impassible.

Makes writing pathfinding algorithms so much more fun. ;)

farmerjumperdon

2006-Mar-22, 04:17 PM

well what about a blackhole?

if that streaches space-time, what is the shortest distance between two points in space on opposite sides of the blackhole?

Now you've gone and done it. You've played the black hole card.

This should spark some interesting debate; I think the solution is still a line.

Let's say I drew a map of the area, placing the black hole on the middle of the page, and the 2 points out near the edge of the page. Now I draw a line between the 2 points. That line is my answer. But that might be seen as cheating since I'm just modeling my solution and not putting it to test in the real world. Though I may come back to my airliner-thru-the-Earth example and say just because it would be difficult (or impossible) to travel doesn't mean it is not the solution.

So in a real world scenario, the shortest distance between 2 points that have a black hole between them would be the straightest possible path that remains just outside the event horizon of the black hole? I suggest this because obviously if the path crossed the event horizon, it would never go anywhere else, much less make it to the other point.

The shortest distance between two points is under construction.

Roy Batty

2006-Mar-22, 04:32 PM

Simple, it's a Rhino Charge (http://www.brianlucas.ca/travel/rhinocharge/) :)

WaxRubiks

2006-Mar-22, 04:42 PM

So in a real world scenario, the shortest distance between 2 points that have a black hole between them would be the straightest possible path that remains just outside the event horizon of the black hole? I suggest this because obviously if the path crossed the event horizon, it would never go anywhere else, much less make it to the other point.

I don't think that the shortest route needs to be travelable to count, so it could go past the event horizon.

farmerjumperdon

2006-Mar-22, 04:44 PM

The shortest distance between two points is under construction.

Ain't that the truth! In the 10 years I've been commuting 37 miles to work, at least some portion of the 27 miles of interstate has ALWAYS been torn up. There has never been a time, not even one single day, when the entire 27 miles was devoid of construction.

jfribrg

2006-Mar-22, 04:55 PM

The shortest distance between two points depends entirely on the distance function being used. In order for a function d(x,y) to qualify as a distance function, the following must be true:

d(x,y) >= 0

d(x,y) = d(y,x)

d(x,y)=0 iff x=y

d(x,y) + d(y,z) >= d(x,z)

the familiar cartesion distance function d(x,y) = sqrt(x^2 + y^2) meets these criteria, but so do many other functions. Tell me which function you are using ,and the answer is easy

WaxRubiks

2006-Mar-22, 05:25 PM

What about, the longest,straight, distance between two points?

:inane smile:

SeanF

2006-Mar-22, 05:34 PM

The shortest distance between two points depends entirely on the distance function being used.

If you define "distance" as being the result of a specific function, then doesn't that mean that there's only one distance between any two points, anyway?

Disinfo Agent

2006-Mar-22, 05:38 PM

If you define "distance" as being the result of a specific function, then doesn't that mean that there's only one distance between any two points, anyway?That's the usal meaning of 'distance', when you think about it. I guess people mangle the terminology because of the phrase 'shortest path'.

Moose

2006-Mar-22, 05:40 PM

What about, the longest,straight, distance between two points?

:inane smile:

The very same as the shortest. :D

Carnifex

2006-Mar-22, 07:29 PM

Shortest path is ALWAYS a straight line. Now the quickest path is quite another story ;)

Disinfo Agent

2006-Mar-22, 07:39 PM

Shortest path is ALWAYS a straight line.But what is a straight line?

jfribrg

2006-Mar-22, 07:39 PM

If you define "distance" as being the result of a specific function, then doesn't that mean that there's only one distance between any two points, anyway?

Yes. The quick answer is that for a given metric space, there is exactly one distance between any two points. However, when you talk about distance (http://en.wikipedia.org/wiki/Distance), you need to specify the metric space (http://en.wikipedia.org/wiki/Metric_space) that you are using. A metric space is defined as some set and a distance function that takes as input two elements of the set and returns a non-negative real number. I defined the distance function in my previous post. Many times the set being used is obvious (usually some n-dimensional crossproduct of the set of reals which is called a cartesian space) and isnt explicitly stated. Also, unless you specify otherwise, you can assume that you are dealing with a cartesian space using the cartesian distance function. When you start dealing with other metric spaces, such as curved surfaces or manifolds (spacetime warping falls into this category), you need to be explicit about the set and the distance function. Any introductory book on topology can give you more information about metric spaces than you would care to know.

BTW, the cartesian distance function is not the only function that works in cartesian space. The Manhattan distance (http://en.wikipedia.org/wiki/Manhattan_distance) function is another example.

jfribrg

2006-Mar-22, 07:43 PM

But what is a straight line?

A straight line between two points a,b is the set of points x such that the distance between a and x plus the distance between x and b is equal to the distance between a and b:

d(a,b) = d(a,x) + d(x,b)

Disinfo Agent

2006-Mar-22, 07:46 PM

Which means it depends on the particular metric space considered.

Glom

2006-Mar-22, 07:58 PM

The shortest distance between two point on the surface of a sphere is a great circle. Flying the Great Circle uses less fuel than drilling a straight line through the mantle.

WaxRubiks

2006-Mar-22, 08:03 PM

a straight line, is the shortest path between two points. wala!

Roy Batty

2006-Mar-22, 08:07 PM

But just where is the instigator of all this huh? Vaelroth? Com'on lets organise a posse & hunt him down to be accountable! :D

Carnifex

2006-Mar-22, 08:21 PM

777 geek - again, it's not the shortest, it's the fastest and the most economic ;)

jfribrg

2006-Mar-22, 08:31 PM

Which means it depends on the particular metric space considered.

Exactly.

jfribrg

2006-Mar-22, 08:32 PM

On the number line, what is the distance between 1.0000... and 0.999999....? (runs and hides before anyone can throw a tomato at him ).

Disinfo Agent

2006-Mar-22, 08:33 PM

Which number line? :razz:

jfribrg

2006-Mar-22, 08:35 PM

Which number line? :razz:

The shortest straight one we can find.

Roy Batty

2006-Mar-22, 08:36 PM

Throwing a big, fat, 0 Tomato at him now :)

Robert Andersson

2006-Mar-22, 08:53 PM

My guess: ca 1.61624 x 10-35 m, but what do I know? ;)

WaxRubiks

2006-Mar-22, 08:55 PM

depend on who is surving.

epenguin

2006-Mar-22, 10:11 PM

The shortest distance between two points depends entirely on the distance function being used. In order for a function d(x,y) to qualify as a distance function, the following must be true:

d(x,y) >= 0

d(x,y) = d(y,x)

d(x,y)=0 iff x=y

d(x,y) + d(y,z) >= d(x,z)

the familiar cartesion distance function d(x,y) = sqrt(x^2 + y^2) meets these criteria, but so do many other functions. Tell me which function you are using ,and the answer is easy

Yeah that's exactly what I said.

Don't mess around, just go straight there without stopping at any other points on the way, don't even go through any points on the way if you can help it.

Fr. Wayne

2006-Mar-23, 06:29 AM

Frog March- Excellent Avatar! I must repeat whatever you define the two points to be (even sub-atomic quanta), the SHORTEST MUST be one point. That it can be as distant as the two extreme edges of the universe does not change the question. Every other distance is longer than one point, because in order for it to be less than one point apart, the two points (however defined) are no longer distinct. Resistance is pointless. (giggle.)

jumbo

2006-Mar-23, 09:50 AM

Im wondering why a geodesic is 'not quite right'

It is locally the shortest path between 2 points in a given metric space as its the 'straightest' possible line between points in that space. This straight line in a curved space being a curve and in a flat space being a line. A straight line is a geodesic in certain spaces.

Im not sure why its not the answer according to Vaelroth.

epenguin

2006-Mar-23, 09:53 AM

...the SHORTEST MUST be one point...)

But surely 'one point' is not a distance? And the distance between a point and itself may be zero [d(x,y)=0 iff x=y as jfribrg says] by arbitrary definition of a metric space but it would not be true in some other space or in everyday life. For example, if the set of points includes one in Parva Craphampton and you are not in Parva Craphampton and you go there and come back to exactly the point where you are/were the distance you have travelled is not zero. We can glimpse a system in which the distance of one point is actually a maximum not a minimum!

Vaelroth excluded zero as an answer.

Actually it's not clear to me whether this was a joke, whether Valeroth wants the answer or whether he or she knows the answer and wants to see if we do. However, I think it is now becoming anti-ecological that I and some of the second-best minds in the world should be occupied with this thing when we could be occupied with something more worthy like solving world hunger or playing the woowoo game, so I THINK WE SHOULD BE TOLD.:)

jfribrg

2006-Mar-23, 01:40 PM

For example, if the set of points includes one in Parva Craphampton and you are not in Parva Craphampton and you go there and come back to exactly the point where you are/were the distance you have travelled is not zero.

In this example you are really changing the metric space. Instead of the set of 3D cartesian points (whether you use 3D cartesian space or a manifold is irrelevant in this example), by talking about going and coming you implicitly add a time dimension to the space, so you have a 4D metric space and a different distance function that incorporates time into the calculation. Your 3D starting and ending points are the same, but because the time is not the same, they are different points in the 4D metric space. If you dont want to include time, then the distance between your starting and ending point is 0. There is nothing in the distance metric that incorporates movement.

epenguin

2006-Mar-23, 04:48 PM

I agree I'm changing the metric space or it's not a metric space anymore. Needn't be time at all - e.g. just the loop sum of distances.

HenrikOlsen

2006-Mar-30, 02:57 PM

A straight line is the shortest distance between two points.

The thing to remember is where you're watching from.

If you're watching from somewhere not on the line it may look curved if the space it's in is bent, but as you traverse the line there will be no curvature seen, so seen from within the possibly bent space it's a straight line.

epenguin

2006-Apr-09, 10:50 PM

When he asked Valeroth seemed to know the answer. He told us what it was not. Which not everyone seems to agree or even noticed. But I think we should now be put out of our misery and be told. At least tell us why ask that qestion

Roy Batty

2006-Apr-09, 11:09 PM

I notice it was his last post here on BAUT... maybe 'real life' has got in the way:think:

epenguin

2006-Apr-10, 12:05 AM

I notice it was his last post here on BAUT... maybe 'real life' has got in the way:think:

What a tragic loss to Science! :cry: In the year 2400 histories may recount how for the whole of the twenty-first and twenty-second centuries people kept on believing a straight line was the shortest distance between two points because the one person who knew...sensational historical discovery in obscure website etc. etc.

WaxRubiks

2006-Apr-10, 12:30 AM

maybe he fell off a cliff following a satnav's instructions going between two points.

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