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Negative pedal curve

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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. 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

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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

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For a parametrically defined curve, its negative pedal curve with pedal point (0; 0) is defined as:[2]

Examples

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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

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The negative pedal curve of a pedal curve with the same pedal point is the original curve.[3]

See also

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

References

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  1. ^ 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
  2. ^ Weisstein, Eric W. "Negative Pedal Curve". mathworld.wolfram.com. Retrieved 2025-06-10.
  3. ^ Edwards, Joseph (1892). An Elementary Treatise On The Differential Calculus (2nd ed.). p. 165.