The catenary evolute is the evolute of a catenary. For 
and real 
, the catenary has parametric
equations
 |  |  |
(1)
|
 |  |  |
(2)
|
giving its evolute as
 |  |  |
(3)
|
 |  |  |
(4)
|
The evolute has a cusp at 
when 
. For 
, the arc length measured
from this cusp, the curvature,
and the tangential angle (for increasing 
and measured from the positive x-axis)
are
 |  |  |
(5)
|
 |  |  |
(6)
|
 |  |  |
(7)
|
where 
denotes the Gudermannian. The tangent
line has slope 
.
