## Point-Point Distance--3-D

The Line Element is

 (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)