Jacobi (elliptic) coordinates

Since

$$\text{cn}^2 u+\text{sn}^2 u=1,$$

$$x=r\ \text{cn}\ u,$$

$$y=r\ \text{sn}\ u,$$

where sn and cn are sinus amplitudinis and cosinus amplitudinis,

yields 

$$x^2 + y^2 =r^2.$$

Using Jacobi's ellipse, we can now construct the following integral. 

$$w=\int_0^u \frac{d\mu}{\sqrt{1-l^2\ \text{sn}^2 \mu}},$$

which satisfies the differential equation associated with the algebraic curve

$$y^2=x^6 + ax^2 + bx + c$$