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
[edit]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]
Parameterization
[edit]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'}}}](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)
Examples
[edit]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]
Properties
[edit]The negative pedal curve of a pedal curve with the same pedal point is the original curve.[3]
See also
[edit]- Fish curve, the negative pedal curve of an ellipse with squared eccentricity 1/2
References
[edit]- ^ a b Lockwood, E. H., ed. (2010) [1961], "Negative Pedals", Book of Curves, Cambridge: Cambridge University Press, pp. 157–160, ISBN 978-0-521-04444-8, retrieved 2025-06-10
- ^ Weisstein, Eric W. "Negative Pedal Curve". mathworld.wolfram.com. Retrieved 2025-06-10.
- ^ Edwards, Joseph (1892). An Elementary Treatise On The Differential Calculus (2nd ed.). p. 165.