TOPICS
Search

Cornoid

DOWNLOAD Mathematica NotebookDownload Wolfram Notebook
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 and 0<=t<2pi.

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

(OEIS A141108 for the decimal expansion of s/a), 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 loop 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 inside the outer envelope and outside the two loops is

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

See also

Sextic Curve

Portions of this entry contributed by Margherita Barile

Explore with Wolfram|Alpha

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

Cornoid

Cite this as:

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

Subject classifications