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On a euclidean plane, the shortest distance between any two distinct points is the line segment joining them. How can I see why this is true?

ReekMaths
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    Relevant: http://en.wikipedia.org/wiki/Triangle_inequality – Austin Mohr Aug 10 '12 at 17:51
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    The definition of the length of a smooth curve segment as a limit of polygonal approximations easily implies the result. But that is not entirely satisfactory as an explanation. – André Nicolas Aug 10 '12 at 17:58
  • @AndréNicolas: Why do you think this is not satisfactory? How else would you define the length of a curve if not by such approximations? (the definition involving an integral is only justified by such an approximation, it seems to me...) – Sam Aug 10 '12 at 18:24
  • The problem is that the definition of length has "shortest distance is given by line segment" built in, so an argument based on that has a tautologous feel. But of course I do not have an alternate definition of distance! – André Nicolas Aug 10 '12 at 18:28
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    What do you mean by "shortest distance"? – Qiaochu Yuan Aug 10 '12 at 18:34
  • Is there a proof that doesn't involve calculus? I suppose, mathematicians who lived before the invention of calculus knew this fact. What could be the way they proceeded into to prove it? Or, did they take it for granted? – ReekMaths Aug 10 '12 at 18:38
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    When you ask "how can I feel", are you looking for an intuitive grokking of the fact, or do you want a formal proof? In order to give the latter, one needs an explicit definition of what the length of something that is not a line segment is, and the only such definition in general use involves calculus in an essential way. It may be doable to prove without calculus that a line segment has minimal length among all piecewise straight curves joining two points, but then which axioms do you want it proved from? (Euclid seems to take something like this as so obvious it's not even stated). – hmakholm left over Monica Aug 10 '12 at 19:03
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    When I want to feel this is true I imagine taking a small rubber band and stretching it between the two points. –  Aug 11 '12 at 02:09

3 Answers3

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Every now and then it's nice to nuke a mosquito.

Let's assume that the path connecting two points $(a,y(a))$ and $(b,y(b))$ can be expressed as a function, and the curve $C(x)$ is given by $C(x) = (x,y(x))$. Then we will proceed using the Calculus of Variations.

The derivative of $C$ wrt $x$ is $(1, y')$, and the functional we want to minimize is the length of the curve $L = \int \|C'\|dx = \int_a^b\sqrt{1 + y' ^2} dx$. If we take $f(x,y,y') = \sqrt{1 + y'^2}$, we get that $\frac{df}{dy} = 0, \frac{df}{dy'} = \frac{y'}{\sqrt{1 + y'^2}}$. Then the Euler-Lagrange equation, sometimes referred to as the fundamental equation of the Calculus of Variations, says exactly that $\dfrac{d}{dx} \left( \frac{y'}{\sqrt{1 + y'^2}}\right) = 0$, which is exactly that $y'$ is a constant.

Thus, if the path connecting the two points is expressible as a function, then the shortest such path is given by a straight line.

EDIT I was certain that someone was in the middle of writing an answer when I typed my tongue-in-cheek response (as so often happens), but as I now see that there is more to add, allow me to extend my answer

The problem here is that we must first define "distance." In the standard Euclidean Plane, the distance between two points is defined to be the length of the line segment between them. So we can drop the word 'shortest' and say that "The distance between any two distant points is the length of the line segment joining them."

Presumably, you want to know that going along any other path will be at least as long. One way of 'seeing this' is that you can approximate any curve with a polygonal path, and these satisfy the triangle inequality, which will make the path longer.

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davidlowryduda
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Think of all possible paths as geometrical figures. In a triangle, the sum of any 2 sides is greater than the 3rd side. Hence if you go by any other path other than a straight line, you travel more.

Rahul Shah
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Let $\gamma(s)$ be a continuous curve in the plane with end-points $\gamma(0) = a$ and $\gamma(1) = b$. Using the Euler-Lagrange equations, the only stationary solution is $\gamma(s) = bs + (1 - s)a$, which is a line connecting the two end-points. See also this.

user02138
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