Shortest Distance Between Two Points is a Straight Line: How?

Factual Questions
1

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?

2

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.

3

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?

4

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.

5

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.

6

Going to start next fall. Took a statistics based math program for my first Bachelor’s so I’ve never had a Calc class.

7

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).

8

It is if you’re a New York cab driver: Taxicab Metric

9

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.

10

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-

11

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 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…

12

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.

13

Yeah, Manhattan distance is distance as described in the OP, as if moving up and across through a series of New York blocks.

14

Yeah, that subscript would’ve helped, wouldn’t it? Mea culpa.

15

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.

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.

16

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.

17

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…

18

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.

19

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.

20

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.