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Trott Curve

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TrottCurve

The Trott curve is a simple example of a quartic curve having 28 real bitangents-the maximum possible for a quartic curve-given by the equation

12^2(x^4+y^4)-15^2(x^2+y^2)+350x^2y^2+81=0
(1)

(Trott 1997).

The four ovals have total area

A=(625Pi(-(49)/(576)|(49)/(62))-576K(7/(sqrt(62))))/(8sqrt(62))
(2)
=0.6881590087...
(3)

(E. Weisstein, Sep. 23, 2026). Here K(k) is the complete elliptic integral of the first kind with elliptic modulus k, while Pi(n|m) is the complete elliptic integral of the third kind with elliptic characteristic n and parameter m. Each oval has area A/4.

The area moment of inertia tensor of the region enclosed by all four ovals, about the origin, is

J=(25[17856E(7/(sqrt(62)))-183744K(7/(sqrt(62)))+195631Pi(-(49)/(576)|(49)/(62))])/(326656sqrt(62))I
(4)
=(0.2625574221...)I
(5)

(E. Weisstein, Sep. 24, 2026), where I is the 2×2 identity matrix and E(k) is the complete elliptic integral of the second kind with elliptic modulus k.

See also

Area, Area Moment of Inertia, Bitangent, Complete Elliptic Integral of the First Kind, Complete Elliptic Integral of the Third Kind, Quartic Curve

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References

Trott, M. "Applying GroebnerBasis to Three Problems in Geometry." Mathematica Educ. Res. 6, 15-28, 1997.

Referenced on Wolfram|Alpha

Trott Curve

Cite this as:

Weisstein, Eric W. "Trott Curve." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/TrottCurve.html

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