Please bear with me, and don't get upset if i have lack in knowledge about spacetime.
Brachistochrone: Given two points A and B in a vertical plane, what is the curve traced out by a point acted on only by gravity, which starts at A and reaches B in the shortest time.
So from my understanding light moves in a 'straight' way (it moves in all ways but the outer paths cancel out?). And it always takes the fastest route. And when light 'bends' around a massive object it actually is still moving in a straight line on a geodesic, which is the fastest route.
So is it fair to say that a brachistochrone is a straight line in the curved space, as its the shortest time path from a to b under a constant gravitational force?