Negative Pedal Curve

Given a curve C and O=(x_0,y_0) a fixed point called the pedal point, then for a point P on C, draw a line perpendicular to OP. The envelope of these lines as P describes the curve C is the negative pedal of C. It can be constructed by considering the perpendicular line segment ((x_1,y_1),(x_2,y_2)) for a curve C parameterized by (f,g). Since one end of the perpendicular corresponds to the point P, (x_1,y_1)=(f,g). Another end point can be found by taking the perpendicular to the OP line, giving




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




Solving the simultaneous equations U(x,y,t)=0 and partialU/partialt=0 then gives the equations of the negative pedal curve as


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

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

See also

Contrapedal Curve, Pedal Curve

Explore with Wolfram|Alpha


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.

Subject classifications