Negative pedal curve
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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
![{\displaystyle X[x,y]={\frac {(y^{2}-x^{2})y'+2xyx'}{xy'-yx'}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a636cd000052812297015f5a109adce9f192d81)
![{\displaystyle Y[x,y]={\frac {(x^{2}-y^{2})x'+2xyy'}{xy'-yx'}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/87d2e638b9b983188b6866ca4b0536ef310583b2)
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