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Trefoil Curve


TrefoilCurve

The "trefoil" curve is the name given by Cundy and Rollett (1989, p. 72) to the quartic plane curve given by the equation

 x^4+x^2y^2+y^4=x(x^2-y^2).
(1)

As such, it is a simply a modification of Kepler's folium with a=1 and b=2

 x^4+2x^2y^2+y^4=2x(x^2-y^2)
(2)

obtained by dropping the coefficients 2.

The area enclosed by the trefoil curve is

 A=(a^2pi)/(4sqrt(3)),
(3)

the geometric centroid (x^_,y^_) of the enclosed region is

x^_=1/2a
(4)
y^_=0
(5)

and the area moment of inertia elements by

I_(xx)=pi/(192sqrt(3))
(6)
I_(xy)=0
(7)
I_(yy)=pi/(12sqrt(3))
(8)

(E. Weisstein, Feb 3, 2018).


See also

Fish Curve, Kepler's Folium, Talbot's Curve, Trefoil Knot, Trifolium

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References

Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 72, 1989.

Cite this as:

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

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