Parabola Catacaustic
ParabolaCatacaustic
The catacaustic of a parabola (t,t^2) opening upward is complicated for a general radiant point (x,y). However, the equations simplify substantially in the case x=infty (i.e., with rays perpendicular to the axis of the parabola), giving
x = 3/2t-2t^3
(1)
y = 3t^2.
(2)
Making the substitution t^'=3t/2 yields
x = 9/4t^'(1-3t^('2))
(3)
y = (27)/4t^('2),
(4)
which is a translated and rotated Tschirnhausen cubic with a=9/4.
If the radiant point is taken at y=infty (i.e., with rays parallel to the axis of the parabola), then the catacaustic degenerates to the single point (0,1/4) as expected since the parabola has a focus at this point.
See also
Catacaustic, Parabola, Tschirnhausen CubicExplore 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
Parabola CatacausticCite this as:
Weisstein, Eric W. "Parabola Catacaustic." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ParabolaCatacaustic.html