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
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
For a parametrically defined curve, its negative pedal curve with pedal point (0; 0) is defined as:[2]
Examples
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] The negative pedal curve of a parabola with respect to its focus is the Tschirnhausen cubic.[3]
Properties
The negative pedal curve of a pedal curve with the same pedal point is the original curve.[4]
See also
- Fish curve, the negative pedal curve of an ellipse with squared eccentricity 1/2
References
- ↑ 1.0 1.1 Lockwood, E. H., ed (2010). "Negative Pedals". Book of Curves. Cambridge: Cambridge University Press. pp. 157–160. ISBN 978-0-521-04444-8. https://www.cambridge.org/core/books/book-of-curves/negative-pedals/3F3257272309E7A6385D790253604A28. Retrieved 2025-06-10.
- ↑ Weisstein, Eric W.. "Negative Pedal Curve" (in en). https://mathworld.wolfram.com/NegativePedalCurve.html.
- ↑ "Tschirnhaus' Cubic". MacTutor History of Mathematics archive. https://mathshistory.st-andrews.ac.uk/Curves/Tschirnhaus/.
- ↑ Edwards, Joseph (1892) (in en). An Elementary Treatise On The Differential Calculus (2nd ed.). pp. 165. http://archive.org/details/in.ernet.dli.2015.109607.
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