Negative pedal curve

In geometry, a negative pedal curve is a plane curve that can be constructed from another plane curve C and a fixed point P. For each point X ≠ P on the curve C, the negative pedal curve has a tangent that passes through X and is perpendicular to line XP. Constructing the negative pedal curve is the inverse operation to constructing a pedal curve.

Definition

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.

Parameterization

For a parametrically defined curve, its negative pedal curve with pedal point is defined as:

Examples

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. The negative pedal curve of a parabola with respect to its focus is the Tschirnhausen cubic.

Properties

The negative pedal curve of a pedal curve with the same pedal point is the original curve.