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
and
so the EulerLagrange Differential Equations become
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) 
See also PointPoint Distance1D, PointPoint Distance2D,
PointQuadratic Distance
© 19969 Eric W. Weisstein
19990525