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Scarabaeus Curve

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Scarabaeus

The Scarabaeus curve is a sextic curve given by the equation

(x^2+y^2)(x^2+y^2+ax)^2-b^2(x^2-y^2)^2=0
(1)

and by the polar equation

r=bcos(2theta)-acostheta,
(2)

where a,b!=0.

The signed area of the standard traversal 0<=theta<=2pi is

A_s=1/2int_0^(2pi)r^2dtheta
(3)
=pi/2(a^2+b^2).
(4)

Because the curve can have overlapping or nested loops, this value weighted by the contour winding number need not equal the area of the geometric union of its bounded regions.

See also

Area, Polar Curve, Signed Area

Portions of this entry contributed by Margherita Barile

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References

Shikin, E. V. Handbook and Atlas of Curves. Boca Raton, FL: CRC Press, pp. 308-311, 1995.

Referenced on Wolfram|Alpha

Scarabaeus Curve

Cite this as:

Weisstein, Eric W., with contributions by Margherita Barile. "Scarabaeus Curve." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ScarabaeusCurve.html

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