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?
The shortest distance between any two distinct points is the line segment joining them.How can I see why this is true?
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$\begingroup$
calculus
general-topology
geometry
euclidean-geometry
analytic-geometry
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8Relevant: http://en.wikipedia.org/wiki/Triangle_inequality – 2012-08-10
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1The *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. – 2012-08-10
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0@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...) – 2012-08-10
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0The 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! – 2012-08-10
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2What do you mean by "shortest distance"? – 2012-08-10
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0Is 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? – 2012-08-10
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4When 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). – 2012-08-10
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4When I want to feel this is true I imagine taking a small rubber band and stretching it between the two points. – 2012-08-11