Negative pedal curve

Circle — negative pedal curve of a limaçon

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

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

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

Properties

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

External links

Negative pedal curve on Mathworld

Differential transforms of plane curves

Unary operations	Evolute Involute Dual curve Inverse curve Parallel curve Isoptic

Unary operations defined by a point	Pedal & Contrapedal curves Negative pedal curve Pursuit curve Caustic

Unary operations defined by two points	Strophoid

Binary operations defined by a point	Roulette

Operations on a family of curves	Envelope

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