next curve  previous curve  2D curves  3D curves  surfaces  fractals  polyhedra 
PLÜCKER QUARTIC, AMPERSAND CURVE
Curve studied by Plücker in 1839.
Julius Plücker (18011868): German mathematician and physicist. The name "ampersand curve" was given by [Cundy and Rolett]. Websites: perso.univrennes1.fr/christophe.ritzenthaler/cours/ellipticcurvecourse.pdf (p. 43) library.msri.org/books/Book35/files/gray.pdf (p 122) 
In 1839, J. Plücker constructed a quartic for which the 28 bitangents are real and distinct.
Equation: . The figure on the right shows the 4 lines y=x, y=x, x=1, x=3/2, and the circle (x1)^{2}+y^{2}=1 that were used to determine this curve, as well as the curve in the case where the constant is zero (rational quartic with 3 double points). It is this curve that Cundy and Rolett call "ampersand curve". 

For cte >0 small enough, the quartic has four components, each having a concave part, called "meniscus", each having a bitangent; added to the 6 times 4 = 24 bitangents linking two components, this indeed makes 28 bitangents in total. 

See the Salmon quartic for a simpler curve with 28 bitangents.
next curve  previous curve  2D curves  3D curves  surfaces  fractals  polyhedra 
© Robert FERRÉOL 2017