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
| (x_2,y_2)=(f,g)+(-(g-y_0),f-x_0), |
(1)
|
or
Plugging into the two-point form of a line then gives
or
Solving the simultaneous equations U(x,y,t)=0 and partialU/partialt= then gives the equations of the negative pedal curve as
![画像:-([(g-y_0)^2-(f-x_0)f]g^'+(2f-x_0)(g-y_0)f^')/((f-x_0)g^'-(g-y_0)f^')](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/NegativePedalCurve/Inline25.svg){kind=link}
![画像:([(f-x_0)^2-(g-y_0)g]f^'+(2g-y_0)(f-x_0)g^')/((f-x_0)g^'-(g-y_0)f^').](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/NegativePedalCurve/Inline28.svg){kind=link}
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 CurveExplore with Wolfram|Alpha
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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 CurveCite this as:
Weisstein, Eric W. "Negative Pedal Curve." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/NegativePedalCurve.html