Cardioid Pedal Curve
In general, the pedal curve of the cardioid is a slightly complicated function.
The pedal curve of the cardioid with respect to the center of its conchoidal circle is the limaçon trisectrix (Ferréol).
CardioidPedal
For the special pedal point of the cardioid cusp, the pedal curve of the cardioid
x = a(1+cost)cost
(1)
y = a(1+cost)sint,
(2)
is
x_p = 2cos^4(1/2t)(2cost-1)
(3)
y_p = 2cos^3(1/2t)sin(3/2t),
(4)
which is Cayley's sextic (Gray 1997, pp. 119-120).
See also
Cardioid, Cardioid Negative Pedal Curve, Cayley's Sextic, Limaçon Trisectrix, Pedal CurveExplore with Wolfram|Alpha
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References
Ferréol, R. "Limaçon Trisectrix." https://mathcurve.com/courbes2d.gb/limacon/limacontrisecteur.shtml.Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, 1997.Referenced on Wolfram|Alpha
Cardioid Pedal CurveCite this as:
Weisstein, Eric W. "Cardioid Pedal Curve." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CardioidPedalCurve.html