Cornoid
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 CurveThis entry contributed by Margherita Barile
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References
Shikin, E. V. Handbook and Atlas of Curves. Boca Raton, FL: CRC Press, p. 134, 1995.Sloane, N. J. A. Sequence A141108 in "The On-Line Encyclopedia of Integer Sequences."Referenced on Wolfram|Alpha
CornoidCite this as:
Barile, Margherita. "Cornoid." From MathWorld--A Wolfram Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/Cornoid.html