Note:

J L On Earth, or rather any planet, finding the shortest time for an object to slide down a slide starting from point A' which is at height h above point A (A', A and the center of Earth are collinear) and reaching the surface at point B. This slide is a Brachistochrone, but it is in this context with varying gravity and the brachistochrone being spread out over above (and over technically) the sphere. No need for the slide to go around Earth more than 1x. No friction involved. And by having it done as simply as possible, (e.g. no Euler-Lagrange equation to find the curve, and using Snell's Law instead, if it makes sense in the context ), multivariable calculus is OK! The drawing is inaccurat for illustration purpose only