haardiameter

Haardiameter is a scale-free geometric descriptor for finite point sets in Euclidean space. It is defined as the mean pairwise distance among the points, normalized by the diameter of the set.

Definition: Let S = {x1, x2, ..., xm} be a finite subset of Euclidean space, with diam(S) = max_{i,j} dist(xi, xj). The haardiameter of S is HD(S) = [1 / C(m, 2)] ∑_{i<j} dist(xi, xj) / diam(S), where C(m, 2) is the number of unordered pairs. HD(S) lies in the interval [0, 1] and is undefined if diam(S) = 0. The measure is invariant under uniform scaling: multiplying all coordinates by a positive factor a leaves HD(S) unchanged.

Properties: HD(S) = 1 when all pairwise distances in S equal the diameter (for example, two-point sets or vertices of a regular simplex). HD(S) = 0 only when all points coincide. In general, HD(S) reflects how spread out the points are relative to the overall diameter; higer values indicate more uniform spreading, while lower values indicate clustering or alignment.

Computation notes: For a finite set, computing HD requires at least all pairwise distances, giving a time complexity on the order of O(m^2). For large or streaming data, sampling or incremental approximations can yield practical estimates.

Applications: Haardiameter is used in pattern recognition and clustering to compare the shape or dispersion of datasets independent of scale. It provides a single-valued descriptor of spread that complements the traditional diameter.

Example: A two-point set at distance D has HD = 1. A three-point equidistant set (an equilateral triangle with side D) also yields HD = 1. A linearly clustered set with varying interpoint gaps typically gives 0 < HD < 1.

Origin: The term haardiameter is used here as a defined concept; it is not drawn from a standard existing metric. See also diameter, mean distance, and shape descriptors.