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Parabola Pedal Curve


ParabolaPedalDirectrix
ParabolaPedalFoot
ParabolaPedalFocusRefl
ParabolaPedalVertex
ParabolaPedalFocus

The pedal curve of the parabola with parametric equations

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

with pedal point (x_0,y_0) is

x_p=((x_0-a)t^2+y_0t)/(t^2+1)
(3)
y_p=(at^3+x_0t+y_0)/(t^2+1).
(4)

On the conic section directrix, the pedal curve of a parabola is a strophoid (top left). On the foot of the conic section directrix, it is a right strophoid (top middle). On reflection of the focus in the conic section directrix, it is a Maclaurin trisectrix (top right). On the parabola vertex, it is a cissoid of Diocles (bottom left; Gray 1997, p. 119). On the focus, it is a straight line (bottom right; Hilbert and Cohn-Vossen 1999, pp. 26-27). On the symmetry axis for a parabola with a=1, it is a conchoid of de Sluze (H. Smith, pers. comm., Aug. 4, 2004). The following table summarizes these special cases.


See also

Parabola, Parabola Negative Pedal Curve, Pedal Curve

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References

Ameseder, A. "Ueber Fusspunktcurven der Kegelschnitte." Archiv Math. u. Phys. 64, 143-144, 1879.Ameseder, A. "Zur Theorie der Fusspunktencurven der Kegelschnitte." Archiv Math. u. Phys. 64, 145-163, 1879.Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, 1997.Hilbert, D. and Cohn-Vossen, S. Geometry and the Imagination. New York: Chelsea, 1999.Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 94-97, 1972.

Referenced on Wolfram|Alpha

Parabola Pedal Curve

Cite this as:

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

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