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 e^2>1/2 (Lockwood 1967, p. 157). It has four cusps and two ordinary double points. For an ellipse with parametric equations
Talbot's curve has parametric equations
where
| c=sqrt(a^2-b^2) |
(9)
|
is the distance between the ellipse center and one of its foci and
| [画像: e=sqrt(1-(b^2)/(a^2))=c/a ] |
(10)
|
is the eccentricity.
The special case a=b 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
where K(k) is a complete elliptic integral of the first kind with elliptic modulus e.
The curvature and tangential angle are
![画像:(4sqrt(2)a^2b^2)/([a^2+b^2+c^2cos(2t)]^(3/2)[a^2+b^2-3c^2cos(2t)])](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/TalbotsCurve/Inline37.svg){kind=link}
See also
Burleigh's Oval, Ellipse, Ellipse Negative Pedal Curve, Ellipse Parallel Curves, Fish Curve, Negative Pedal Curve, Trefoil CurveExplore with Wolfram|Alpha
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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." https://mathshistory.st-andrews.ac.uk/Curves/Talbots/.Referenced on Wolfram|Alpha
Talbot's CurveCite this as:
Weisstein, Eric W. "Talbot's Curve." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/TalbotsCurve.html