In Euclidean space , the curve that minimizes the distance between two points is clearly a straight line segment. This can be shown mathematically as follows using calculus of variations and the so-called Euler-Lagrange differential equation. The line element in is given by
(1)
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so the arc length between the points and is
(2)
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and the quantity we are minimizing is
(3)
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Finding the derivatives gives
(4)
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(5)
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and
(6)
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(7)
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so the Euler-Lagrange differential equations become
(8)
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(9)
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These give
(10)
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(11)
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Taking the ratio,
(12)
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(13)
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(14)
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which gives
(15)
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(16)
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Therefore, and , so the solution is
(17)
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which is the parametric representation of a straight line with parameter . Verifying the arc length gives
(18)
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where
(19)
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(20)
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