Determining explicitly the Mordell-Weil group of certain rational elliptic surfaces

Let $A,B$ be nonzero rational numbers. Consider the elliptic curve $E_{A,B}\mathrm{/}\mathbb{Q}(t)$ with Weierstrass equation $y^2=x^3+A{t}^6+B$. An algorithm to determine $\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}{E}_{A,B}(\mathbb{Q}(t))$ as a function of $(A,B)$ was presented in a recent paper by Desjardins and Naskrecki. We will give a different and shorter proof for the correctness of that algorithm, using a more geometric approach and discuss for which classes of examples this approach might be useful.