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PLÜCKER QUARTIC, AMPERSAND CURVE

Curve studied by Plücker in 1839.
Julius Plücker (1801-1868): German mathematician and physicist.
The name "ampersand curve" was given by [Cundy and Rolett].
Websites:
perso.univ-rennes1.fr/christophe.ritzenthaler/cours/elliptic-curve-course.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:
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The figure on the right shows the 4 lines y=x, y=-x, x=1, x=3/2, and the circle (x-1)2+y2=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.
 
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© Robert FERRÉOL 2017