Plane algebraic curve
The lemniscate of Gerono In algebraic geometry , the lemniscate of Gerono , or lemniscate of Huygens , or figure-eight curve , is a plane algebraic curve of degree four and genus zero and is a lemniscate curve shaped like an ∞ {\displaystyle \infty } symbol, or figure eight. It has equation
x 4 − x 2 + y 2 = 0. {\displaystyle x^{4}-x^{2}+y^{2}=0.} It was studied by Camille-Christophe Gerono .
Parameterization Because the curve is of genus zero, it can be parametrized by rational functions; one means of doing that is
x = t 2 − 1 t 2 + 1 , y = 2 t ( t 2 − 1 ) ( t 2 + 1 ) 2 . {\displaystyle x={\frac {t^{2}-1}{t^{2}+1}},\ y={\frac {2t(t^{2}-1)}{(t^{2}+1)^{2}}}.} Another representation is
x = cos φ , y = sin φ cos φ = sin ( 2 φ ) / 2 {\displaystyle x=\cos \varphi ,\ y=\sin \varphi \,\cos \varphi =\sin(2\varphi )/2} which reveals that this lemniscate is a special case of a Lissajous figure .
Dual curve The dual curve (see Plücker formula ), pictured below, has therefore a somewhat different character. Its equation is
( x 2 − y 2 ) 3 + 8 y 4 + 20 x 2 y 2 − x 4 − 16 y 2 = 0. {\displaystyle (x^{2}-y^{2})^{3}+8y^{4}+20x^{2}y^{2}-x^{4}-16y^{2}=0.} Dual to the lemniscate of Gerono
References
External links Wikimedia Commons has media related to Lemniscate of Gerono .