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]

This image shows an ampersand curve on the Cartesian plane.

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.

Notes

1. "Mathematical Curves" (PDF). abel.math.harvard.edu.
2. Cundy, Rollett (1981). Mathematical Models. Tarquin Publications. ISBN 9780906212202.
3. "Ampersand Curve". www.statisticshowto.com. 29 December 2021.
4. "Ampersand Curve Genus". people.math.carleton.ca.
5. "Ampersand Curve History". mathcurve.com.

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

Further reading

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