Point-Point Distance--3-Dimensional

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)

so the arc length between the points and is

(2)

and the quantity we are minimizing is

(3)

Finding the derivatives gives

			(4)
			(5)

and

			(6)
			(7)

so the Euler-Lagrange differential equations become

			(8)
			(9)

These give

(10)

(11)

Taking the ratio,

(12)

(13)

(14)

which gives

(15)

(16)

Therefore, and , so the solution is

(17)

which is the parametric representation of a straight line with parameter . Verifying the arc length gives

(18)

where

(19)

(20)