In some applications, selecting points that align with the coordinate axis may be undesirable. Therefore, extensive sampling theory has been developed to determine methods that avoid alignments while distributing the points uniformly. In sampling-based motion planning, grids sometimes yield unexpected behavior because a row of points may align nicely with a corridor in . In some cases, a solution is obtained with surprisingly few samples, and in others, too many samples are necessary. These alignment problems, when they exist, generally drive the variance higher in computation times because it is difficult to predict when they will help or hurt. This provides motivation for developing sampling techniques that try to reduce this sensitivity.
Discrepancy theory and its corresponding sampling methods were developed to avoid these problems for numerical integration [738]. Let be a measure space, such as . Let be a collection of subsets of that is called a range space. In most cases, is chosen as the set of all axis-aligned rectangular subsets; hence, this will be assumed from this point onward. With respect to a particular point set, , and range space, , the discrepancy [965] for samples is defined as (see Figure 5.7)