nearaxis

Nearaxis is a term used in geometry and data analysis to describe a relationship between a set of points or a path and a reference line, emphasizing small deviations from that line. It is applied in contexts where an object or sequence approximately follows a linear path, or where a central axis is used as a model for alignment.

Formally, let l be a line in Euclidean space and S a set of points. S is said to be nearaxis with respect to l if the maximum perpendicular distance from any point in S to l is bounded by a tolerance epsilon > 0. The nearaxis deviation d(S, l) = max_{p in S} dist(p, l) quantifies this closeness. For a polyline or trajectory, nearaxis analysis studies how well the path stays within a narrow corridor around l, and it can be extended to three dimensions with a central axis represented by a line.

Variants of nearaxis analysis include near-axis regression and near-axis line fitting, which seek a line that minimizes the nearaxis deviation for a given dataset. Computational approaches often adapt standard line-fitting or robust regression techniques to optimize distance-based loss functions, possibly incorporating weighting or outlier handling.

Applications of nearaxis thinking appear in trajectory simplification, handwriting recognition, road network analysis, structural monitoring, and quality control where an approximately linear pattern is expected. The term is not a standard mathematical term with a canonical definition in major references; instead, it appears in niche discussions to describe proximity to a central axis. As such, it is best understood as a descriptive concept related to measuring and minimizing distance to a line.

See also: distance to a line, line fitting, principal component analysis, axis alignment, tolerances.