Bicorn
The bicorn, sometimes also called the "cocked hat curve" (Cundy and Rollett 1989, p. 72), is the name of a collection of quartic curves studied by Sylvester in 1864 and Cayley in 1867 (MacTutor Archive). The bicorn is given by the parametric equations
(Lawrence 1972, p. 147) and Cartesian equation
| y^2(a^2-x^2)=(x^2+2ay-a^2)^2 |
(3)
|
(Lawrence 1972, p. 147; Cundy and Rollett 1989, p. 72; Mactutor, with the final a corrected to be squared instead of to the first power).
The bicorn has cusps at (+/-a,0).
The area enclosed by the curve is
| A=1/3(16sqrt(3)-27)pia^2. |
(4)
|
The curvature and tangential angle are given by
![画像: kappa(t)=(6sqrt(2)(cost-2)^3(3cost-2)sect)/(a[73-80cost+9cos(2t)]^(3/2)) phi(t)=1/6 +(tan^(-1)[(sqrt(7)-4)tan(1/2t)]-tan^(-1)[(sqrt(7)+4)tan(1/2t)])/(6t)](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/Bicorn/NumberedEquation3.svg){kind=link}
for -pi<t<pi. There does not seem to be a simple closed form for the arc length of the curve, but its numerical value is approximately given by 5.0565300a.
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References
Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 72, 1989.Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 147-149, 1972.MacTutor History of Mathematics Archive. "Bicorn." https://mathshistory.st-andrews.ac.uk/Curves/Bicorn/.Smith, D. E. History of Mathematics, Vol. 2: Special Topics of Elementary Mathematics. New York: Dover, p. 327, 1958.Referenced on Wolfram|Alpha
BicornCite this as:
Weisstein, Eric W. "Bicorn." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Bicorn.html