TOPICS
Natural Logarithm Catacaustic
The catacaustic of the natural logarithm 
specified parametrically as
 |  |  |
(1)
|
 |  |  |
(2)
|
is a complicated expression for an arbitrary radiant point.
However, for a point 
, the catacaustic
becomes
 |  |  |
(3)
|
 |  |  |
(4)
|
Making the substitution 
then gives the equivalent parametrization
 |  |  |
(5)
|
 |  |  |
(6)
|
which is the equation of a catenary.
See also
Catacaustic, Catenary, Natural LogarithmExplore with Wolfram|Alpha
References
Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, p. 207, 1972.Referenced on Wolfram|Alpha
Natural Logarithm CatacausticCite this as:
Weisstein, Eric W. "Natural Logarithm Catacaustic." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/NaturalLogarithmCatacaustic.html