Ampersand curve

In geometry, the ampersand curve is a type of quartic plane curve. It is named after its resemblance to the ampersand symbol by Henry Cundy and Arthur Rollett.^[1]^[2]

It is described as:

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

The graph on the cartesian plane 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 a variation of the Plücker quartic with the following equation:

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

The ampersand curve follows the equation when k=0.

The curve has 6 real horizontal tangents at: $({\frac {1}{2}},\pm {\frac {\surd 5}{2}}),({\frac {159-\surd 201}{120}},\pm {\frac {\surd {1389+67\surd {\frac {67}{3}}}}{140}}),$ and $({\frac {159+\surd 201}{120}},\pm {\frac {\surd {1389-67\surd {\frac {67}{3}}}}{140}})$

And 4 real vertical tangents at: $(-{\frac {1}{10}},\pm {\frac {\surd {23}}{10}})$ and $({\frac {3}{2}},{\frac {\surd {3}}{2}}).$

It is only one of a few curves that have no value of x in its domain with only one y value.

Notes

^ "Mathematical Curves" (PDF). abel.math.harvard.edu.
^ Cundy, Rollett (1981). Mathematical Models. Tarquin Publications. ISBN 9780906212202.
^ "Ampersand Curve". www.statisticshowto.com. 29 December 2021.
^ "Ampersand Curve Genus". people.math.carleton.ca.
^ "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]

Ampersand curve

Notes

References

Further reading