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

Let $A,B$ be nonzero rational numbers. Consider the elliptic curve EA,B/Q(t) with Weierstrass equation y²=x³+At⁶+B. An algorithm to determine rank EA,B(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.