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Talbot's Curve

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TalbotsCurve

A curve investigated by Talbot which is the ellipse negative pedal curve with respect to the ellipse's center for ellipses with eccentricity e^2>1/2 (Lockwood 1967, p. 157). It has four cusps and two ordinary double points. For an ellipse with parametric equations

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

Talbot's curve has parametric equations

x=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=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.

The special case a=b gives a circle.

TalbotsCurveParallels

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

The area and arc length are

A=((10a^2b^2-a^4-b^4)pi)/(8ab)
(11)
s=4bK(e),
(12)

where K(k) is a complete elliptic integral of the first kind with elliptic modulus e.

The curvature and tangential angle are

kappa(t)=(4sqrt(2)a^2b^2)/([a^2+b^2+c^2cos(2t)]^(3/2)[a^2+b^2-3c^2cos(2t)])
(13)
phi(t)=tan^(-1)((btant)/a).
(14)

See also

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

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