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Ellipse Pedal Curve

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The pedal curve of an ellipse with parametric equations

x=acost
(1)
y=bsint
(2)

and pedal point (x_0,y_0) is given by

f=(a[ax_0sin^2t+bcost(b-y_0sint)])/(b^2cos^2t+a^2sin^2t)
(3)
g=(b[by_0cos^2t+asint(a-x_0cost)])/(b^2cos^2t+a^2sin^2t).
(4)
EllipsePedalFocus

The pedal curve of an ellipse with pedal point at the focus is a circle (Hilbert and Cohn-Vossen 1999, pp. 25-26).

EllipsePedal

For other pedal points, the pedal curves are more complicated, as illustrated above.

See also

Ellipse, Ellipse 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.Hilbert, D. and Cohn-Vossen, S. Geometry and the Imagination. New York: Chelsea, 1999.

Cite this as:

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

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