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# Negative Pedal Curve

Given a curve and a fixed point called the pedal point, then for a point on , draw a line perpendicular to . The envelope of these lines as describes the curve is the negative pedal of . It can be constructed by considering the perpendicular line segment for a curve parameterized by . Since one end of the perpendicular corresponds to the point , . Another end point can be found by taking the perpendicular to the line, giving

 (1)

or

 (2) (3)

Plugging into the two-point form of a line then gives

 (4)

or

 (5)

Solving the simultaneous equations and then gives the equations of the negative pedal curve as

 (6) (7)

If a curve is the pedal curve of a curve , then is the negative pedal curve of (Lawrence 1972, pp. 47-48).

The following table summarizes the negative pedal curves for some common curves.

 curve pedal point negative pedal curve cardioid negative pedal curve origin circle cardioid negative pedal curve point opposite cusp cissoid of Diocles circle negative pedal curve inside the circle ellipse circle negative pedal curve outside the circle hyperbola ellipse negative pedal curve with center Talbot's curve ellipse negative pedal curve with focus ovoid ellipse negative pedal curve with focus two-cusped curve line any point parabola parabola negative pedal curve origin semicubical parabola parabola negative pedal curve focus Tschirnhausen cubic

Contrapedal Curve, Pedal Curve

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## References

Ameseder, A. "Theorie der negativen Fusspunktencurven." Archiv Math. u. Phys. 64, 164-169, 1879.Ameseder, A. "Negative Fusspunktcurven der Kegelschnitte." Archiv Math. u. Phys. 64, 170-176, 1879.Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 46-49, 1972.Lockwood, E. H. "Negative Pedals." Ch. 19 in A Book of Curves. Cambridge, England: Cambridge University Press, pp. 156-159, 1967.

## Referenced on Wolfram|Alpha

Negative Pedal Curve

## Cite this as:

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