Ampersand curve

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.^[1][2]

The ampersand curve is the graph of the equation

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

The graph of the ampersand curve has three crunode points where it intersects itself at (0,0), (1,1), and (1,-1).^[3] The curve has a genus of 0.^[4]

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.^[5]

It is the special case of the Plücker quartic

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

with ${\displaystyle k=0.}$

The curve has 6 real horizontal tangents at

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

And 4 real vertical tangents at ${\displaystyle \left(-{\tfrac {1}{10}},\pm {\tfrac {\sqrt {23}}{10}}\right)}$ and ${\displaystyle \left({\tfrac {3}{2}},{\tfrac {\sqrt {3}}{2}}\right).}$

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

• 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, [1]
• Frost, Percival, Elementary treatise on curve tracing, 1960, [2]

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