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Cornoid

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Cornoid

The cornoid is the curve illustrated above given by the parametric equations

x=acost(1-2sin^2t)
(1)
y=asint(1+2cos^2t),
(2)

where a>0.

It is a sextic algebraic curve with equation

-4a^6+3a^2x^4+x^6+8a^4y^2-6a^2x^2y^2+3x^4y^2-5a^2y^4+3x^2y^4+y^6=0.
(3)

The arc length of the curve is given by

s=4[E(k)-3K(k)+3Pi(1/4,k)]
(4)
=10.6017029...
(5)

(OEIS A141108), where K(k) is a complete elliptic integral of the first kind, E(k) is a complete elliptic integral of the second kind, Pi(z,k) is a complete elliptic integral of the third kind, and k=sqrt(2)i.

The area of a single of the loops is

A_(loop)=1/8a^2(3pi-8),
(6)

the area of the outer envelope is

A_(envelope)=1/4a^2(3pi+8),
(7)

and the area of the region enclosed is

A_(enclosed)=A_(enclosed)-2A_(loop)
(8)
=4a^2.
(9)

See also

Sextic Curve

This entry contributed by Margherita Barile

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References

Sloane, N. J. A. Sequence A141108 in "The On-Line Encyclopedia of Integer Sequences."Shikin, E. V. Handbook and Atlas of Curves. Boca Raton, FL: CRC Press, p. 134, 1995.

Cite this as:

Barile, Margherita. "Cornoid." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/Cornoid.html

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