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
The graph of the ampersand curve has three crunode points where it intersects itself at , , and . 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
with .
The curve has six real horizontal tangents at
and four real vertical tangents at
It is an example of a curve that has no value of x in its domain with only one y value.
Notes
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