I am currently dealing with the question of which numbers can be expressed as a sum of two rational cubes ($n=x^3+y^3$, $x,y\in\mathbb{Q}$). I've already learned that I can understand the equation $n=x^3+y^3$ as an elliptic curve (given by Weierstraß-form $y^2=x^3+Ax+B$) and try to find rational points on this curve.
My question is: How can the equations $n=x^3+y^3$ and $y^2=x^3+Ax+B$ be transformed into each other?
And in what way does projective geometry play a role here?
P.S.: I've started to read the Elliptic Curve Handbook by Conwell(1999) and Taxicabs and Sums of Two Cubes by Silverman (1993), but it feels like I need some rough intuitive introduction before being able to understand it completely. Thanks!