the starting point (point \#1), then for both arcs use a $2:3$ split of the associated variance budget and work out the gaussian distribution of the corresponding intermediate point’s position. Sub-division of the arc of length 2 is straightforward, while the length 3 arc would first be split $1:2$ using the same approach.

Rather than choosing starting points randomly in space and then determining the location of the halfway-round-the-loop point, it makes sense to perform stratified spatial sampling on a lattice. To do so, we choose the first point to be the lattice point and take the halfway point to be gaussian distributed with mean the first point and standard deviation a characteristic length. We then continue sampling a loop of that length and later scale it around the midpoint between the first point and the halfway point appropriately to obtain a unit loop. In that way, the midpoint is gaussian distributed around the lattice point with a given length scale.

If we want unbiased integration by taking loops for each lattice point, the lattice points have to be representative for loops sampled in their vicinity, with a characteristic length being that of the grid. This is certainly true if 