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GeezerVison is showing mathworld.wolfram.com — Ellipse Negative Pedal Curve -- from Wolfram MathWorld

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Ellipse Negative Pedal Curve

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EllipseNegativePedalCurveOrigin
Ellipse negative pedal curve with respect to the origin

For an ellipse with parametric equations

x=acost
(1)
y=bsint,
(2)

the negative pedal curve with respect to the origin has parametric equations

x_n=acos^3t+((2a^2-b^2)costsin^2t)/a
(3)
=((a^2+c^2sin^2t)cost)/a
(4)
=acost(1+e^2sin^2t)
(5)
y_n=bsin^3t+((2b^2-a^2)sintcos^2t)/b
(6)
=((a^2-2c^2+c^2sin^2t)sint)/b
(7)
=(asint(1-2e^2+e^2sin^2t))/(sqrt(1-e^2)),
(8)

where

c=sqrt(a^2-b^2)
(9)

is the distance between the ellipse center and one of its foci and

e=sqrt(1-(b^2)/(a^2))=c/a
(10)

is the eccentricity. For b/a=1, the base curve is a circle, whose negative pedal curve with respect to the origin is also a circle. For sqrt(2)/2<b/a<1, the curve becomes a "squashed" ellipse. For 0<b/a<sqrt(2)/2, the curve has four cusps and two ordinary double points and is known as Talbot's curve (Lockwood 1967, p. 157).

EllipseNegativePedalCurveFocus
Ellipse negative pedal curve with respect to the focus

Taking the pedal point at a focus (i.e., (x,y)=(c,0)) gives the negative pedal curve

x_n=acost-csin^2t
(11)
y_n=((a^2-2c^2+accost)sint)/b.
(12)

Lockwood (1957) terms this family of curves Burleigh's ovals. The curve is a circle for e=0 and a smooth ovoid for 0<e<1/2. At e=1/2 the parametric equations first have a vanishing derivative. For 1/2<e<1 the curve has two cusps and an ordinary double point.

The special case of the negative pedal curve for pedal point (x,y)=(c,0) and e^2=1/2 (i.e., b/a=sqrt(2)/2) is here dubbed the fish curve.

See also

Burleigh's Oval, Circle Negative Pedal Curve, Ellipse, Ellipse Pedal Curve, Fish Curve, Negative Pedal Curve, Talbot's Curve

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References

Ameseder, A. "Negative Fusspunktcurven der Kegelschnitte." Archiv Math. u. Phys. 64, 170-176, 1879.Hilton, H. Plane Algebraic Curves. Oxford, England: Oxford University Press, p. 64, 1932.Lockwood, E. H. "Negative Pedal Curve of the Ellipse with Respect to a Focus." Math. Gaz. 41, 254-257, 1957.Lockwood, E. H. A Book of Curves. Cambridge, England: Cambridge University Press, p. 157, 1967.MacTutor History of Mathematics Archive. "Talbot's Curve." https://mathshistory.st-andrews.ac.uk/Curves/Talbots/.Salmon, G. A Treatise on Higher Plane Curves, Intended as a Sequel to a Treatise on Conic Sections, 3rd ed. Dublin, Ireland: Hodges, p. 107, 1879.

Referenced on Wolfram|Alpha

Ellipse Negative Pedal Curve

Cite this as:

Weisstein, Eric W. "Ellipse Negative Pedal Curve." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/EllipseNegativePedalCurve.html

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