Negative pedal curve

In the plane, for every point X other than P there is a unique line through X perpendicular to XP. For a given curve in the plane and a given fixed point P, called the pedal point, the negative pedal curve is the envelope of the lines XP for which X lies on the given curve.^[1]

For a parametrically defined curve, its negative pedal curve with pedal point (0; 0) is defined as:^[2]

${\displaystyle X[x,y]={\frac {(y^{2}-x^{2})y'+2xyx'}{xy'-yx'}}}$

${\displaystyle Y[x,y]={\frac {(x^{2}-y^{2})x'+2xyy'}{xy'-yx'}}}$

The negative pedal curve of a line is a parabola. The negative pedal curves of a circle are an ellipse if P is chosen to be inside the circle, and a hyperbola if P is chosen to be outside the circle.^[1] The negative pedal curve of a parabola with respect to its focus is the Tschirnhausen cubic.^[3]