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# Talbot's Curve

A curve investigated by Talbot which is the ellipse negative pedal curve with respect to the ellipse's center for ellipses with eccentricity (Lockwood 1967, p. 157). It has four cusps and two ordinary double points. For an ellipse with parametric equations

 (1) (2)

Talbot's curve has parametric equations

 (3) (4) (5) (6) (7) (8)

where

 (9)

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

 (10)

is the eccentricity.

The special case gives a circle.

The curve is also very similar in appearance to ellipse parallel curves (Arnold 1990, p. x).

The area and arc length are

 (11) (12)

The curvature and tangential angle are

 (13) (14)

Burleigh's Oval, Ellipse, Ellipse Negative Pedal Curve, Ellipse Parallel Curves, Fish Curve, Negative Pedal Curve, Trefoil Curve

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## References

Arnold, V. I. Singularities of Caustics and Wave Fronts. Dordrecht, Netherlands: Kluwer, 1990.Lockwood, E. H. A Book of Curves. Cambridge, England: Cambridge University Press, p. 157, 1967.MacTutor History of Mathematics Archive. "Talbot's Curve." http://www-groups.dcs.st-and.ac.uk/~history/Curves/Talbots.html.

## Cite this as:

Weisstein, Eric W. "Talbot's Curve." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/TalbotsCurve.html