Bicorn

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title: "Bicorn" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["plane-curves", "quartic-curves"] description: "Mathematical curve with two cusps" topic_path: "general/plane-curves" source: "https://en.wikipedia.org/wiki/Bicorn" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Mathematical curve with two cusps ::

::figure[src="https://upload.wikimedia.org/wikipedia/commons/5/54/Bicorn.svg" caption="Bicorn"] ::

In geometry, the bicorn, also known as a cocked hat curve due to its resemblance to a bicorne, is a rational quartic curve defined by the equation y^2 \left(a^2 - x^2\right) = \left(x^2 + 2ay - a^2\right)^2. It has two cusps and is symmetric about the y-axis.

History

In 1864, James Joseph Sylvester studied the curve y^4 - xy^3 - 8xy^2 + 36x^2y+ 16x^2 -27x^3 = 0 in connection with the classification of quintic equations; he named the curve a bicorn because it has two cusps. This curve was further studied by Arthur Cayley in 1867.

Properties

::figure[src="https://upload.wikimedia.org/wikipedia/commons/6/6c/Bicorn-inf.jpg" caption="A transformed bicorn with ''a'' = 1"] ::

The bicorn is a plane algebraic curve of degree four and genus zero. It has two cusp singularities in the real plane, and a double point in the complex projective plane at (x=0, z=0). If we move x=0 and z=0 to the origin and perform an imaginary rotation on x by substituting ix/z for x and 1/z for y in the bicorn curve, we obtain \left(x^2 - 2az + a^2 z^2\right)^2 = x^2 + a^2 z^2. This curve, a limaçon, has an ordinary double point at the origin, and two nodes in the complex plane, at x= \pm i and z=1.

The parametric equations of a bicorn curve are \begin{align} x &= a \sin\theta \ y &= a , \frac{(2 + \cos\theta) \cos^2\theta}{3 + \sin^2\theta} \end{align} with -\pi \le \theta \le \pi.

References

References

1. Lawrence, J. Dennis. (1972). "A catalog of special plane curves". Dover Publications.
2. "Bicorn". mathcurve.
3. (1908). "The Collected Mathematical Papers of James Joseph Sylvester". Cambridge University press.
4. "Bicorn". The MacTutor History of Mathematics.

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