Capproximal

Capproximal is a term used in geometry and data analysis to describe a relation between points relative to a common reference point, indicating that the two distances are close to each other within a multiplicative tolerance. Specifically, in a metric space (X,d), two points p and q are called c-approximal with respect to a reference point r ∈ X if the distances d(p,r) and d(q,r) satisfy d(p,r)/d(q,r) ∈ [1/c, c] for some constant c ≥ 1. Equivalently, max{d(p,r), d(q,r)} ≤ c · min{d(p,r), d(q,r)}. The parameter c is called the approximate factor, with c = 1 meaning exact equality of distances.

Notes: The relation is symmetric in p and q but not necessarily transitive. It can be used to describe local symmetry around a center, or to select pairs of points that lie in roughly the same concentric band around r.

Examples:

- In the plane, points p = (0,0), q = (2,0) and r = (1,0) have d(p,r) = d(q,r) = 1, so they are 1-approximal w.r.t r.

- Points p = (0,0), q = (3,0) with r = (1,0) have distances 1 and 2, which are within a factor 2, so they are 2-approximal.

Applications include clustering based on radial shells, robust proximity queries, and spatial indexing where approximate concentricity matters.

Etymology: formed from approximate and proximal, with c denoting a constant tolerance. See also proximal, approximate, metric spaces.