Order of magnitude comparisons of distance

Order of magnitude reasoning - reasoning by rough comparisons of the sizes of quantities - is often called 'back of the envelope calculation', with the implication that the calculations are quick though approximate. This paper exhibits an interesting class of constraint sets in which order of magnitude reasoning is demonstrably fast. Specifically, we present a polynomial-time algorithm that can solve a set of constraints of the form 'Points a and b are much closer together than points c and d'. We prove that this algorithm can be applied if 'much closer together' is interpreted either as referring to an infinite difference in scale or as referring to a finite difference in scale, as long as the difference in scale is greater than the number of variables in the constraint sets. We also prove that the first-order theory over such constraints is decidable.