Given a straight segment of track of length , add a small segment so that
the track bows into a circular Arc. Find the maximum displacement of the bowed
track. The Pythagorean Theorem gives

(1) 
But is simply , so

(2) 
Solving (1) and (2) for gives

(3) 
Expressing the length of the Arc in terms of the central angle,
But is given by

(5) 
so plugging in gives

(6) 

(7) 
For ,

(8) 
Therefore,
Keeping only terms to order ,

(10) 

(11) 
so

(12) 
and

(13) 
If we take
and 1 foot, then
feet.
© 19969 Eric W. Weisstein
19990525