Ampersand curve

The ampersand curve on the Cartesian plane.

In geometry, the ampersand curve is a type of quartic plane curve. It was named after its resemblance to the ampersand symbol by Henry Cundy and Arthur Rollett.

The ampersand curve is the graph of the equation

$6x^{4}+4y^{4}-21x^{3}+6x^{2}y^{2}+19x^{2}-11xy^{2}-3y^{2}=0.$

The graph of the ampersand curve has three crunode points where it intersects itself at (0,0), (1,1), and (1,−1). The curve has a genus of 0.

The curve was originally constructed by Julius Plücker as a quartic plane curve that has 28 real bitangents, the maximum possible for bitangents of a quartic.

It is the special case of the Plücker quartic

$(x+y)(y-x)(x-1)\left(x-{\tfrac {3}{2}}\right)-2\left(y^{2}+x(x-2)\right)^{2}-k=0,$

with k = 0.

The curve has six real horizontal tangents at

$\left({\frac {1}{2}},\pm {\frac {\sqrt {5}}{2}}\right),\qquad \left({\frac {159-{\sqrt {201}}}{120}},\pm {\frac {\sqrt {1389+67{\sqrt {\frac {67}{3}}}}}{40}}\right),\qquad \left({\frac {159+{\sqrt {201}}}{120}},\pm {\frac {\sqrt {1389-67{\sqrt {\frac {67}{3}}}}}{40}}\right),$

and four real vertical tangents at

$\left(-{\frac {1}{10}},\pm {\frac {\sqrt {23}}{10}}\right),\qquad \left({\frac {3}{2}},{\frac {\sqrt {3}}{2}}\right).$

It is an example of a curve that has no value of x in its domain with only one y value.

References

• Piene, Ragni, Cordian Riener, and Boris Shapiro. "Return of the plane evolute." Annales de l'Institut Fourier. 2023
• Figure 2 in Kohn, Kathlén, et al. "Adjoints and canonical forms of polypols." Documenta Mathematica 30.2 (2025): 275-346.
• Julius Plücker, Theorie der algebraischen Curven, 1839,
• Frost, Percival, Elementary treatise on curve tracing, 1960,

Further reading

• "Plücker's Quartic". mathworld.wolfram.com.
• "Ampersand Curve Points". mathworld.wolfram.com.