What is the equation of the Bicorn?

Cartesian equation: y^2(a^2 - x^2) = (x^2 + 2ay - a^2)^2; Parametric equations: x = a sin t, y = (a cos^2 t (2 + cos t))/(3 + sin^2 t). a is the half-width: the cusps sit at (±a, 0).

Classic Cartesian curves

Bicorn

The cocked-hat curve y²(a² − x²) = (x² + 2ay − a²)²: two arcs meeting in sharp cusps at (±a, 0).

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Formulas and key values

Cartesian equation: $y^2(a^2 - x^2) = (x^2 + 2ay - a^2)^2$
Parametric equations: $x = a\sin t$ $y = \frac{a\cos^2 t\,(2 + \cos t)}{3 + \sin^2 t}$
Cusps: $(\pm a,\ 0)$

a is the half-width: the cusps sit at (±a, 0).

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01 · Definition

What is the Bicorn?

The bicorn, or cocked-hat curve, is a rational quartic studied by James Joseph Sylvester in 1864 and Arthur Cayley in 1867. Its outline is exactly what the name suggests: a hat with two points, one arc on top and one below, meeting in cusps at (±a, 0).

Being rational, it has a parametrisation by ordinary functions, x = a·sin t and y = a·cos²t·(2 + cos t)/(3 + sin²t). The upper arc crosses the y-axis at height a, the lower one at a/3.

02 · Properties

Key properties

Cusps at (±a, 0).
Crosses the y-axis at y = a and y = a/3.
Rational parametrisation: x = a·sin t, y = a·cos²t·(2 + cos t)/(3 + sin²t).
Symmetric about the y-axis.
A quartic studied by Sylvester (1864) and Cayley (1867).

03 · Questions

Bicorn: common questions

What is the equation of the Bicorn?: Cartesian equation: y^{^2}(a^{^2} - x^{^2}) = (x^{^2} + 2ay - a^{^2})^{^2}; Parametric equations: x = a sin t, y = (a cos^{^2} t (2 + cos t))/(3 + sin^{^2} t). a is the half-width: the cusps sit at (±a, 0).