Lemniskata

Skoči na: navigacija, iskanje

Lemniskata

Lemniskata je implicitno podana krivulja s formulo:

$\! (x^2+y^2)^2 = 2(x^2-y^2)$

Pretvorba

$x \rightarrow r \cos \varphi$

$y \rightarrow r \sin \varphi$

Polarno

$r^4 = 2(r^2 \cos ^2 \varphi - r^2 \sin \varphi )$

$r^2 = 2(\cos ^2 \varphi - \sin \varphi) = 2 \cos 2 \varphi$

$0 \le 2\varphi \le \frac{\pi}{2} \Rightarrow 0 \le \varphi \le \frac{\pi}{4}$

$\frac{5\pi}{2} \le 2 \varphi \le 2\pi \Rightarrow \frac{5\pi}{4} \le \varphi \le \frac{\pi}{4}$

$\frac{7\pi}{2} \le 2 \varphi \le \frac{9\pi}{2} \Rightarrow \frac{7\pi}{4} \le \varphi \le \frac{9\pi}{4}$