first path to the reverse of the second solves the problem of finding a path with the correct distribution between two given points. One easily sees that the distribution of the midpoint is still gaussian (being the product of two gaussian distributions). Hence, we can sample a loop by recursively sampling an intermediate point in the interval between a given start and end point.

This method, known as the “d (‘doubling’) loop algorithm” [@Gies:2005sb], manages to generate closed loops with the desired distribution with very little effort. A slight problem of this approach is that it only generates loops for which the number of vertices is a power of two, but as one usually is only interested in getting below a given resolution, and this method will at worst require an effort too large by a factor 2, this is practically irrelevant. Nevertheless, the reasoning presented above is readily generalized to non-equal subdivisions (e.g. 1/3 plus 2/3), and from there to also allow recursive subdivision into other numbers of parts, but this is typically not needed. In order to generate a loop containing 10 points using the extended d-loop method, one would first determine the opposite point (point \#6) of 