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Limaçon Trisectrix


The limaçon trisectrix is a trisectrix that is a special case of the rose curve with n=1/3 (possibly with translation, rotation, and scaling). It was studied by Archimedes, as well as by Étienne Pascal in 1630.

LimaconTrisectrix

In its most commonly written standard form, the limaçon trisectrix has polar equation

 r=a(1+2costheta)
(1)

(Ferréol). It can be expressed as the Cartesian equation

 a^2(3x^2-y^2)+(x^2+y^2)^2=4ax(x^2+y^2)
(2)

or

 a^2(x^2+y^2)=(x^2+y^2-2ax)^2.
(3)

The limaçon trisectrix is the cardioid pedal curve with respect to the center of its conchoidal circle (Ferréol).

It has arc length

 s=12aE((2sqrt(2))/3),
(4)

where E(k) is a complete elliptic integral of the second kind. Its outer boundary encloses an area

 A=1/2a^2(3sqrt(3)+pi),
(5)

and its inner lopp has area

 A_(loop)=1/2a^2(2pi-3sqrt(3)).
(6)
LimaconTrisectrixRose

As a special case of the rose curve r=cos(ntheta), the polar equation is given by

 r=asin(theta/3)
(7)

which must be rotated by 90 degrees, scaled by a factor of 2, shifted by distance 1 to the right, and plotted from theta=0 to 3pi to obtain the curve in its standard form.


See also

Limaçon, Rose Curve, Trisectrix

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References

Ferréol, R. "Limaçon Trisectrix." https://mathcurve.com/courbes2d.gb/limacon/limacontrisecteur.shtml.Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, p. 175, 1972.

Cite this as:

Weisstein, Eric W. "Limaçon Trisectrix." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LimaconTrisectrix.html

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