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.{{Cite web|url=https://abel.math.harvard.edu/~knill/sofia/data/curves.pdf|website=abel.math.harvard.edu|title=Mathematical Curves}}{{Cite book |last=Cundy, Rollett |title=Mathematical Models |date=1981 |publisher=Tarquin Publications |isbn=9780906212202}}

File:Ampersandcurve.svg

The ampersand curve is the graph of the equation

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

The graph of the ampersand curve has three crunode points where it intersects itself at (0,0), (1,1), and (1,-1).{{Cite web|url=https://www.statisticshowto.com/ampersand-curve/|website=www.statisticshowto.com|title=Ampersand Curve|date=29 December 2021 }} The curve has a genus of 0.{{Cite web|url=https://people.math.carleton.ca/~cingalls/studentProjects/Katie's%20Site/html/Ampersand%20Curve.html|website=people.math.carleton.ca|title=Ampersand Curve Genus}}

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.{{Cite web|url=https://mathcurve.com/courbes2d.gb/quarticdeplucker/quarticdeplucker.shtml|website=mathcurve.com|title=Ampersand Curve History}}

It is the special case of the Plücker quartic

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

with $k=0.$

The curve has 6 real horizontal tangents at

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

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

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

Notes

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, [https://archive.org/details/bub_gb_y-XnP4ghQoIC/page/n276/mode/1up]
Frost, Percival, Elementary treatise on curve tracing, 1960, [https://archive.org/details/elementarytreat00fros/page/n239/mode/1up]

ampersand curve

Notes

References

Further reading