centroid

noun

1. The point at which gravitational force (or other universally and uniformly acting force) may be supposed to act on a given rigid, uniformly dense body; the centre of gravity or centre of mass.

   “For example, if a postal code area (PCA) has 1000 distinct residences, we might suppose all 1000 residences are at the centroid of the PCA. Centroids are commonly used, for example, with geographic information systems and CD-ROM phone books (Francis, Lowe, Rushton and Rayco 1999).”

   “operatorname Centroid(𝒳)=(∫xg(x)dx)/(∫g(x)dx)      (6) where the integrals are taken over the whole space ℝⁿ, and g is the characteristic function of the subset, which is 1 inside 𝒳 and 0 outside it [27].”

2. (specifically)The point of intersection of the three medians of a given triangle; the point whose (Cartesian) coordinates are the arithmetic mean of the coordinates of the three vertices.
3. the point whose (Cartesian) coordinates are the arithmetic mean of the coordinates of a given finite set of points.
4. An analogue of the centre of gravity of a nonuniform body in which local density is replaced by a specified function (which can take negative values) and the place of the body's shape is taken by the function's domain.

   “The centroid of an arbitrary function f is given by #92;frac#123;#92;intxf(x)dx#125;#123;#92;intf(x)dx#125;, where the integrals are calculated over the domain of f.”

5. the arithmetic mean (alternatively, median) position of a cluster of points in a coordinate system based on some application-dependent measure of distance.
6. Given a tree of n nodes, either (1) a unique node whose removal would split the tree into subtrees of fewer than n/2 nodes, or (2) either of a pair of adjacent nodes such that removal of the edge connecting them would split the tree into two subtrees of exactly n/2 nodes.

   “1974 [Prentice-Hall], Narsingh Deo, Graph Theory with Applications to Engineering and Computer Science, 2017, Dover, page 248, Just as in the case of centers of a tree (Section 3-4), it can be shown that every tree has either one centroid or two centroids. It can also be shown that if a tree has two centroids, the centroids are adjacent.”