Draft:Ampersand curve

• Comment: The only problem of this draft is no reliable sources (see WP:RS). Dedhert.Jr (talk) 11:24, 2 May 2025 (UTC)

The ampersand curve is a type of quartic plane curve. It is named after its resemblance to the ampersand symbol by Henry Cundy and and Arthur Rollett.

This image shows an ampersand curve on the Cartesian plane.

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). The curve has a genus of 0.

The curve was originally constructed by Julius Plücker as a quadratic plane curve that has 28 real bitangents.

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.