I ask why adjective 'elliptic' for elliptic curves?
I have read that for the calculus of length of arc of ellipse founds a integral of type $\displaystyle\int_0^a\sqrt{\frac{1-k^2t^2}{1-t^2}} dt$ (elliptic integral).
Then call $u$ the integrand I have $u^2(1-t^2)=1-k^2t^2$.
This curve (in a $t/u$-plane) have a connection with elliptic curves (i.e. with a Weierstrass equation)?
I have also read that elliptic curve on $\mathbb{C}$ can be parametrized with elliptic functions, in particular with Weierstrass elliptic function.
Can someone please synthesize the story in between elliptic curves, elliptic integral and elliptic function?
An elliptic curve admits a parametrization through the Weierstrass $\wp$ function, whose periods are given by elliptic integrals of the first kind. The perimeter of an ellipse is given by an elliptic integral of the second kind, and I believe this is the reason for the adjective elliptic in the description of objects like $\{(x,y)\in\mathbb{C}^2: y^2=x^3+px+q\}$ (with suitable constraints for the discriminant).