# Shortest Distance Between Two Points is a Straight Line: How?
**URL:** https://boards.straightdope.com/t/shortest-distance-between-two-points-is-a-straight-line-how/430796
**Category:** Factual Questions
**Created:** [December 21, 2007, 4:27am UTC](https://boards.straightdope.com/t/shortest-distance-between-two-points-is-a-straight-line-how/430796 "2007-12-21T04:27:52Z")
**Posts on this page:** 20
**Page:** 1
### Author:  [@Cyberhwk](https://boards.straightdope.com/u/Cyberhwk)
#### Post date: [December 21, 2007, 4:27am UTC](https://boards.straightdope.com/t/shortest-distance-between-two-points-is-a-straight-line-how/430796/1 "2007-12-21T04:27:52Z")
I’ll try to explain this as simple as possible. Hopefully someone else can show me where I’m wrong as simply as possible.
Common wisdom tells us that the shortest distance between two points is a straight line. My question is how does this work?
Picture a graph and I need to get from Point ‘A’ at (0,5) to Point ‘B’ at (5,0). If I were to go from (0,5) --\> (0,0) --\> (5,0) my total distance would equal 10. But lets say I take a “more direct” rout. (0,5) --\> (1,5) --\> (1,4) --\> (2,4) … (5,0) going in a zig-zag pattern. My distance traveled is still the same. It’s still 10. Even if I took a huge number of microscopic tiny zig zags the rise will always equal the run. It will always equal 10.
So how come the distance from ‘A’ to ‘B’ is 7.071? What does it stop equaling 10 and start equaling 7.071?
I’m guessing “line” and “infinitely small zig-zag” is comparing apples to oranges, but I’d still expect maybe some point where it starts approaching 7.071. Does it just change instantaneous? Is “zig-zag” approaching “line” just a visual illusion?
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### Author:  [@Snarky\_Kong](https://boards.straightdope.com/u/Snarky_Kong)
#### Post date: [December 21, 2007, 4:33am UTC](https://boards.straightdope.com/t/shortest-distance-between-two-points-is-a-straight-line-how/430796/2 "2007-12-21T04:33:27Z")
Yes, it is just an illusion. Think of “zooming in” on the line. You’ll notice that it’s not approximating a diagonal at all. I’m sure somebody’ll be along shortly with something better to say.
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### Author:  [@Drum\_God](https://boards.straightdope.com/u/Drum_God)
#### Post date: [December 21, 2007, 4:38am UTC](https://boards.straightdope.com/t/shortest-distance-between-two-points-is-a-straight-line-how/430796/3 "2007-12-21T04:38:37Z")
Well, I’m no mathmetician, but I’ll take a stab at this. It’s late here, so forgive me if I’m not very precise.
First of all, the very definition of a line is the shortest distance between two points. Therefore, the answer to your question is contained in the question itself. If it’s not the shortest distance, it’s not a line. If it is the shortest distance, it is a line.
As to your zig-zagging path, you’re creating triangles. Connect the end points of your zig and zag and you’ve got a hypotenuse. Pythagoras teaches us that A^2 + B^2= C^2. Therefore, the hypotenuse is necessarily shorter than the sum of the two legs of the triangle. Add up all the hypotenuses of your zig-zag path and you get the shortest distance between two points. Ergo, you’ve got a line.
Cool?
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### Author:  [@Joey\_P](https://boards.straightdope.com/u/Joey_P)
#### Post date: [December 21, 2007, 5:02am UTC](https://boards.straightdope.com/t/shortest-distance-between-two-points-is-a-straight-line-how/430796/4 "2007-12-21T05:02:49Z")
[QUOTE=Cyberhwk]
I’m guessing “line” and “infinitely small zig-zag” is comparing apples to oranges, but I’d still expect maybe some point where it starts approaching 7.071. Does it just change instantaneous? Is “zig-zag” approaching “line” just a visual illusion?
[/QUOTE]
You never took Calculus did you? The entire concept of calculus is basically contained in what you said, right down to the concept of approaching.
It’s been a loooong time since I’ve done this, so I can’t recall how it was all taught to us, but I believe you’re missing something fundamental here. Off the top of my head I’m going to say it has something to do with the fact that even if you make infinitely small zig zags you’re still going out of your way to get from point A to point B. If you add up the hypotenuses of all those zig zags it should equal your 7.071 number.
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### Author:  [@Bryan\_Ekers](https://boards.straightdope.com/u/Bryan_Ekers)
#### Post date: [December 21, 2007, 5:11am UTC](https://boards.straightdope.com/t/shortest-distance-between-two-points-is-a-straight-line-how/430796/5 "2007-12-21T05:11:26Z")
If you limit yourself to only up-down and left-right movement, you’ll never do better than 10. Conveniently, the universe is not so restricted.
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### Author:  [@Cyberhwk](https://boards.straightdope.com/u/Cyberhwk)
#### Post date: [December 21, 2007, 5:11am UTC](https://boards.straightdope.com/t/shortest-distance-between-two-points-is-a-straight-line-how/430796/6 "2007-12-21T05:11:53Z")
[QUOTE=Joey P]
You never took Calculus did you?
[/QUOTE]
Going to start next fall. Took a statistics based math program for my first Bachelor’s so I’ve never had a Calc class.
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### Author:  [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)
#### Post date: [December 21, 2007, 5:20am UTC](https://boards.straightdope.com/t/shortest-distance-between-two-points-is-a-straight-line-how/430796/7 "2007-12-21T05:20:04Z")
The confusion is that you’re looking at two different definitions of distance: one in which the distance between **x** and **y** is || **x** - **y** ||, and one in which it’s sqrt(( **x** - **y** )[sup]t[/sup]( **x** - **y** )). They’re measuring different things, so why would you expect them to agree?
It is interesting to think about what happens as the step size of your movements parallel to the axes goes to zero. Intuitively your series of zig-zags is approaching a straight line, but I’m not sure that the transition between metrics is continuous (in other words, the distance is 10 as long as you’re not at 0, and 5sqrt(2) when you are).
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### Author:  [@Squink](https://boards.straightdope.com/u/Squink)
#### Post date: [December 21, 2007, 5:21am UTC](https://boards.straightdope.com/t/shortest-distance-between-two-points-is-a-straight-line-how/430796/8 "2007-12-21T05:21:40Z")
[QUOTE=Bryan Ekers]
If you limit yourself to only up-down and left-right movement, you’ll never do better than 10. Conveniently, the universe is not so restricted.
[/QUOTE]
It is if you’re a New York cab driver: [Taxicab Metric](http://mathworld.wolfram.com/TaxicabMetric.html)
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### Author:  [@Dag\_Otto](https://boards.straightdope.com/u/Dag_Otto)
#### Post date: [December 21, 2007, 5:40am UTC](https://boards.straightdope.com/t/shortest-distance-between-two-points-is-a-straight-line-how/430796/9 "2007-12-21T05:40:04Z")
[QUOTE=Cyberhwk]
Going to start next fall. Took a statistics based math program for my first Bachelor’s so I’ve never had a Calc class.
[/QUOTE]
Keep up with the inquisitive nature. You’ll find that ‘the shortest distance between two points is a straight line’ is a special case for a plane surface. On a curved surface, say a globe, the shortest path is a curve. That’s why maps of airplane routes look so odd.
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### Author:  [@Frylock](https://boards.straightdope.com/u/Frylock)
#### Post date: [December 21, 2007, 6:09am UTC](https://boards.straightdope.com/t/shortest-distance-between-two-points-is-a-straight-line-how/430796/10 "2007-12-21T06:09:55Z")
The distance from 5,0 to 0,0 to 0,5 is 10.
The distance from 5,0 to 1,1 to 0,5 is 8.24
The distance from 5,0 to 2,2 to 0,5 is 7.22
The distance from 5,0 to 2.25,2.25 is 7.11
And so on. This series approaches 7.07.
I don’t think this answers your question, but it may shed some light on how to think about it.
-FrL-
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### Author:  [@brad\_d](https://boards.straightdope.com/u/brad_d)
#### Post date: [December 21, 2007, 6:57am UTC](https://boards.straightdope.com/t/shortest-distance-between-two-points-is-a-straight-line-how/430796/11 "2007-12-21T06:57:15Z")
[QUOTE=ultrafilter]
The confusion is that you’re looking at two different definitions of distance: one in which the distance between **x** and **y** is || **x** - **y** ||, and one in which it’s sqrt(( **x** - **y** )[sup]t[/sup]( **x** - **y** )). They’re measuring different things, so why would you expect them to agree?
[/QUOTE]
I’m feeling really foolish - how are those different?
Isn’t || **x** - **y** || the usual notation for the Euclidian norm? If so, what is that but sqrt(( **x** - **y** )[sup]t[/sup]( **x** - **y** ))?
The Wikipedia page on [norms](http://en.wikipedia.org/wiki/Norm_%28mathematics%29) uses || **x** - **y** ||[sub]1[/sub] to represent what is sometimes called the Manhattan norm - is that what you meant in the first instance up there? Otherwise, I’m quite confused…
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### Author:  [@CookingWithGas](https://boards.straightdope.com/u/CookingWithGas)
#### Post date: [December 21, 2007, 11:38am UTC](https://boards.straightdope.com/t/shortest-distance-between-two-points-is-a-straight-line-how/430796/12 "2007-12-21T11:38:04Z")
The OP is describing an oft-discussed paradox, but I can’t remember where I first read it many years ago.
To sum up what others have said, no matter how small you make the zig zags, they are still zig zags. Because points on a plane are infinitely many, you can never get to the point where the stair step becomes so small it can be ignored mathematically.
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### Author:  [@Capt.Ridley\_s\_Shooting\_Party](https://boards.straightdope.com/u/Capt.Ridley_s_Shooting_Party)
#### Post date: [December 21, 2007, 1:13pm UTC](https://boards.straightdope.com/t/shortest-distance-between-two-points-is-a-straight-line-how/430796/13 "2007-12-21T13:13:16Z")
> [@](#):
>
> ||x - y||1 to represent what is sometimes called the Manhattan norm - is that what you meant in the first instance up there?
Yeah, Manhattan distance is distance as described in the OP, as if moving up and across through a series of New York blocks.
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### Author:  [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)
#### Post date: [December 21, 2007, 1:20pm UTC](https://boards.straightdope.com/t/shortest-distance-between-two-points-is-a-straight-line-how/430796/14 "2007-12-21T13:20:02Z")
[QUOTE=brad\_d]
I’m feeling really foolish - how are those different?
Isn’t || **x** - **y** || the usual notation for the Euclidian norm? If so, what is that but sqrt(( **x** - **y** )[sup]t[/sup]( **x** - **y** ))?
The Wikipedia page on [norms](http://en.wikipedia.org/wiki/Norm_%28mathematics%29) uses || **x** - **y** ||[sub]1[/sub] to represent what is sometimes called the Manhattan norm - is that what you meant in the first instance up there? Otherwise, I’m quite confused…
[/QUOTE]
Yeah, that subscript would’ve helped, wouldn’t it? Mea culpa.
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### Author:  [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)
#### Post date: [December 21, 2007, 4:39pm UTC](https://boards.straightdope.com/t/shortest-distance-between-two-points-is-a-straight-line-how/430796/15 "2007-12-21T16:39:57Z")
I’d like to compliment the OP on a good question—and knowing a little calculus just makes the question more intriguing. If the zig-zaggy path is “approaching” a straight line, why isn’t its length approaching the length of a straight line?
[QUOTE=CookingWithGas]
The OP is describing an oft-discussed paradox, but I can’t remember where I first read it many years ago.
[/QUOTE]
And, yes, I vaguely remember seeing something like this before, too, but I can’t remember where.
I think the answer has already been covered pretty well by other posters. Restating it in my own words, if only for my own benefit, it’s that, with every zig-zag, where you’re traveling along the horizontal and vertical sides of a triangle instead of the hypotenuse, you’re deviating from the direct, straight-line course, and whether you make one big deviation or lots and lots of teeny tiny ones, it adds up to the same total of out-of-your-way-ness.
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### Author:  [@msmith537](https://boards.straightdope.com/u/msmith537)
#### Post date: [December 21, 2007, 4:44pm UTC](https://boards.straightdope.com/t/shortest-distance-between-two-points-is-a-straight-line-how/430796/16 "2007-12-21T16:44:57Z")
[QUOTE=Cyberhwk]
Picture a graph and I need to get from Point ‘A’ at (0,5) to Point ‘B’ at (5,0). If I were to go from (0,5) –\> (0,0) –\> (5,0) my total distance would equal 10. But lets say I take a “more direct” rout. (0,5) –\> (1,5) –\> (1,4) –\> (2,4) … (5,0) going in a zig-zag pattern. My distance traveled is still the same. It’s still 10. Even if I took a huge number of microscopic tiny zig zags the rise will always equal the run. It will always equal 10.
[/QUOTE]
The distance between Point A and Point B is 7.07 using Pythagriums theorum. You are adding the bottom and the side of the triangle, not measureing the hypotenuse (the direct route).
If you were walking city blocks then you would be correct. Since you can’t walk diagonally across the blocks, you’re distance would be the same whether you went in a bunch of smaller zig zags or just walked the length of one avenue then up the side street.
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### Author:  [@Una\_Persson](https://boards.straightdope.com/u/Una_Persson)
#### Post date: [December 21, 2007, 6:21pm UTC](https://boards.straightdope.com/t/shortest-distance-between-two-points-is-a-straight-line-how/430796/17 "2007-12-21T18:21:23Z")
I’m remembering something vaguely from Calc 2 - something like the _fastest_ way between two points, under some situation, is an inverted cycloid? Ah, Googling shows me it is the brachistochrone problem I was thinking of…
> **[Cycloid](https://en.wikipedia.org/wiki/Cycloid)**
>
> In geometry, a cycloid is the curve traced by a point on a circle as it rolls along a straight line without slipping. A cycloid is a specific form of trochoid and is an example of a roulette, a curve generated by a curve rolling on another curve.
> The cycloid, with the cusps pointing upward, is the curve of fastest descent under uniform gravity (the brachistochrone curve). It is also the form of a curve for which the period of an object in simple harmonic motion (rolling up and down repetitively)...
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### Author:  [@Gangster\_Octopus](https://boards.straightdope.com/u/Gangster_Octopus)
#### Post date: [December 21, 2007, 6:37pm UTC](https://boards.straightdope.com/t/shortest-distance-between-two-points-is-a-straight-line-how/430796/18 "2007-12-21T18:37:14Z")
[QUOTE=Thudlow Boink]
I’d like to compliment the OP on a good question—and knowing a little calculus just makes the question more intriguing. If the zig-zaggy path is “approaching” a straight line, why isn’t its length approaching the length of a straight line?And, yes, I vaguely remember seeing something like this before, too, but I can’t remember where.
[/QUOTE]
I think , because in fact a zig-zaggy path is NOT approaching a straight line at all. It is in fact approaching an infinite number of points with zero length.
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### Author:  [@merrily](https://boards.straightdope.com/u/merrily)
#### Post date: [December 21, 2007, 6:46pm UTC](https://boards.straightdope.com/t/shortest-distance-between-two-points-is-a-straight-line-how/430796/19 "2007-12-21T18:46:32Z")
Here’s another way to think of it:
Think of Point ‘A’ at (0,5) to the origin (0,0) to Point ‘B’ at (5,0) as a triangle.
‘A’ to origin is 5, 'B; to origin is 5, as you’ve pointed out.
Now make two triangles: ‘A’ to (0,2.5) to (2.5,2.5), and ‘B’ to (2.5, 0) to (2.5,2.5).
Project the horizontal parts of each triangle down on the x-axis, and you will see that they equal (0,0) to B. Do the same with the vertical parts, and they equal (0,0) to A.
No matter how many small right triangles you make where A to B is the sum of the hypotenueses, the horizontals when projected down to the x-axis will always cover (0,0) to (5,0), and similarly for the y-axis. By making smaller triangles, you are essentially taking the exact same length of line and just bending it. That’s why the distance doesn’t change.
The volume changes like crazy, of course.
Now imagine a circle with the center at the origin, which goes through each axis at 5 and -5. What calculus came from is the idea of estimating the length of one quarter of that circle (say, from A to B) by using the first big triangle, and then using two triangles the touch the circle, and then using more and more --the same way that a pentagon fits more tightly inside a circle than a square, and a hexagon better than a pentagon, and each time you are getting closer to the area of a circle, until you have a polygon with so many sides that each side become a point and you’ve got the circle. The way the areas of those shapes as more sides are added converges on the circle’s area is the wonderful insight that led to calculus.
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### Author:  [@Eonwe](https://boards.straightdope.com/u/Eonwe)
#### Post date: [December 21, 2007, 8:44pm UTC](https://boards.straightdope.com/t/shortest-distance-between-two-points-is-a-straight-line-how/430796/20 "2007-12-21T20:44:23Z")
Man. Calc 2 was the math class in which I did the worst in college.
Picture a graph where the Y value is the distance along a path from point A to B. The X axis represents the number of ‘steps’ in your line.
As X approaches infinity, Y stays the same (10), but at infinity, Y is 1.071.
There are many equations that display this same sort of action (as x approaches infinity, the y value approaches some value, but at infinity, the y value is something other than what it was approaching.
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