6.1 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Ampersand curve | 1/1 | https://en.wikipedia.org/wiki/Ampersand_curve | reference | science, encyclopedia | 2026-05-05T12:04:06.292461+00:00 | kb-cron |
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
6
x
4
+
4
y
4
−
21
x
3
+
6
x
2
y
2
+
19
x
2
−
11
x
y
2
−
3
y
2
=
0.
{\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). 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
(
x
+
y
)
(
y
−
x
)
(
x
−
1
)
(
x
−
3
2
)
−
2
(
y
2
+
x
(
x
−
2
)
)
2
−
k
=
0
,
{\displaystyle (x+y)(y-x)(x-1)(x-{\tfrac {3}{2}})-2(y^{2}+x(x-2))^{2}-k=0,}
with
k
=
0.
{\displaystyle k=0.}
The curve has 6 real horizontal tangents at
(
1
2
,
±
5
2
)
,
{\displaystyle \left({\frac {1}{2}},\pm {\frac {\sqrt {5}}{2}}\right),}
(
159
−
201
120
,
±
1389
+
67
67
/
3
40
)
,
{\displaystyle \left({\frac {159-{\sqrt {201}}}{120}},\pm {\frac {\sqrt {1389+67{\sqrt {67/3}}}}{40}}\right),}
and
(
159
+
201
120
,
±
1389
−
67
67
/
3
40
)
.
{\displaystyle \left({\frac {159+{\sqrt {201}}}{120}},\pm {\frac {\sqrt {1389-67{\sqrt {67/3}}}}{40}}\right).}
And 4 real vertical tangents at
(
−
1
10
,
±
23
10
)
{\displaystyle \left(-{\tfrac {1}{10}},\pm {\tfrac {\sqrt {23}}{10}}\right)}
and
(
3
2
,
3
2
)
.
{\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 ==
== 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.