centroido

Centroido is a concept in geometry and data analysis referring to a central point that summarizes the location of a finite or continuous distribution of mass or points. The term is not standard in mainstream references, but is used in some texts to emphasize a generalization of the centroid and center of mass to varied metrics and densities.

For a finite set of points p_i in R^n with weights w_i > 0, centroido is c = (Σ w_i p_i) / (Σ w_i). In Euclidean space with equal weights, c is the arithmetic mean of the coordinates.

For a region R with density function ρ(x) ≥ 0, the centroido is c = (1/M) ∫_R x ρ(x) dx, with M = ∫_R ρ(x) dx. This recovers the ordinary centroid for uniform density.

Properties: In Euclidean space with positive weights the centroido minimizes the sum of squared distances to the mass points; it is unique if weights are positive and the data are not degenerate. It is invariant under rigid motions and scales appropriately with size.

Computation: In practice, for polygons or polyhedra, the centroido can be computed by decomposing into triangles or tetrahedra and using area- or volume-weighted centroids; for large data sets, linear-time algorithms exist.

Applications and relation to other concepts: Used in clustering initialization, computer graphics, and physics. It remains distinct from the graph-theoretic centroid and the geometric median; centroido supports squared-distance optimization, while the geometric median minimizes sum of distances. See also: centroid, geometric median, center of mass.