Natural Logarithm Catacaustic
The catacaustic of the natural logarithm lnx specified parametrically as
x = t
(1)
y = lnt
(2)
is a complicated expression for an arbitrary radiant point.
NaturalLogarithmCatacaustic
However, for a point x->infty, the catacaustic becomes
x_c = [画像:(1+t^2)/(2t)]
(3)
y_c = lnt-1.
(4)
Making the substitution u=lnt then gives the equivalent parametrization
x_c = coshu
(5)
y_c = u-1,
(6)
which is the equation of a catenary.
See also
Catacaustic, Catenary, Natural LogarithmExplore with Wolfram|Alpha
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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