Self-Intersecting spiral moves

You are given a list of n float numbers x_1, x_2, x_3, … x_n, where x_i > 0. A traveler starts at point (0, 0) and moves x_1 metres to the north, then x_2 metres to the west, x_3 to the south, x_4 to the east and so on (after each move his direction changes counter-clockwise).
Write an single-pass algorithm that uses O(1) memory to determine, if the travelers path crosses itself, or not (i.e. if it’s self-intersecting)
2 1 1 2 -> crosses
1 2 3 4 -> doesn’t cross