Negative pedal curve

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.{{Citation |title=Negative Pedals |date=2010 |work=Book of Curves |pages=157–160 |editor-last=Lockwood |editor-first=E. H. |orig-date=1961 |url=https://www.cambridge.org/core/books/book-of-curves/negative-pedals/3F3257272309E7A6385D790253604A28 |access-date=2025-06-10 |place=Cambridge |publisher=Cambridge University Press |isbn=978-0-521-04444-8}}

For a parametrically defined curve, its negative pedal curve with pedal point (0; 0) is defined as:{{Cite web |last=Weisstein |first=Eric W. |title=Negative Pedal Curve |url=https://mathworld.wolfram.com/NegativePedalCurve.html |access-date=2025-06-10 |website=mathworld.wolfram.com |language=en}}

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

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

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.

The negative pedal curve of a pedal curve with the same pedal point is the original curve.{{Cite book |last=Edwards |first=Joseph |url=http://archive.org/details/in.ernet.dli.2015.109607 |title=An Elementary Treatise On The Differential Calculus |date=1892 |edition=2nd |pages=165 |language=en}}

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

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{{Differential transforms of plane curves}}

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Category:Curves

Category:Differential geometry