Ampersand curve

From Wikipedia, the free encyclopedia

The ampersand curve.
The ampersand curve.

In mathematics, the ampersand curve is a quartic plane curve given by the equation:

\ (y^2-x^2)(x-1)(2x-3)=4(x^2+y^2-2x)^2.

It is an algebraic curve of genus zero, with three ordinary double points, all in the real plane. By the Plücker formulas, the dual curve has degree six, with four double points in the real part of the plane corresponding to double tangents of the ampersand curve.

Dual curve of the ampersand curve.
Dual curve of the ampersand curve.

The dual curve has the formula

19(16x2 − 5y2)(3x2 − 19y2)2 − 48048x5 + 146272x4 − 364472y2x3 + 547456x3 + 561664x2 − 726664y2x2 − 501344y2x + 241920x + 252776y4x + 38400 − 114672y2 + 110776y4 = 0.
Languages