# 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* on that curve. 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 (0; 0) is defined as

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

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

## Properties

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

## See also

- Fish curve, the negative pedal curve of an ellipse with squared eccentricity 1/2

## External links

Original source: https://en.wikipedia.org/wiki/Negative pedal curve.
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