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Parabola Negative Pedal Curve

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ParabolaNegativePedalCurve
Parabola negative pedal curve

The parabola negative pedal curve is the negative pedal curve of a parabola with respect to a given pedal point. For a parabola with parametric equations

x=at^2
(1)
y=2at,
(2)

the negative pedal curve for a pedal point (x_0,0) has equation

x_n=(at^2[a(3t^2+4)-x_0])/(at^2+x_0)
(3)
y_n=-(t[a^2t^4-2a(t^2+2)x_0+x_0^2])/(at^2+x_0).
(4)

Taking the pedal point at the origin (x_0,y_0)=0 gives

x_n=a(4+3t^2)
(5)
y_n=-at^3,
(6)

which is a semicubical parabola. Similarly, taking (x_0,y_0)=(a,0) gives

x_n=3at^2
(7)
y_n=at(t^2-3),
(8)

which is a Tschirnhausen cubic, the parabolic member of the ichthyoid curve family (Cundy 2001).

See also

Ichthyoid Curve, Negative Pedal Curve, Parabola, Parabola Pedal Curve, Tschirnhausen Cubic

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References

Ameseder, A. "Negative Fusspunktcurven der Kegelschnitte." Archiv Math. u. Phys. 64, 170-176, 1879.Cundy, H. M. "The Ichthyoids." Math. Gaz. 85, 439-445, 2001. https://doi.org/10.2307/3621749.Lockwood, E. H. A Book of Curves. Cambridge, England: Cambridge University Press, p. 157, 1967.

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Parabola Negative Pedal Curve

Cite this as:

Weisstein, Eric W. "Parabola Negative Pedal Curve." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ParabolaNegativePedalCurve.html

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