Talbot's Curve

A curve investigated by Talbot which is the ellipse negative pedal curve with respect to the ellipse's center for

ellipses witheccentricity (Lockwood 1967, p. 157). It has four cusps and two ordinary double points. For an ellipse withparametric 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 certain parallel curves of an ellipse (Arnold 1990, p. x).

The area and arc length are

(11)

(12)

where is a complete elliptic integral of the first kind with elliptic modulus .

The curvature and tangential angle are

(13)

(14)

SEE ALSO:Burleigh's Oval, Ellipse, Ellipse Negative Pedal Curve, Fish Curve, Negative Pedal Curve, Trefoil Curve 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.http://mathworld.wolfram.com/TalbotsCurve.html