centroid

Etymology
From. From 1844, used as a replacement for the older terms "centre of gravity" and "centre of mass" in situations described in purely geometrical terms, and subsequently used for further generalisations.

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.
 * 2) * 2020, Cheng Zhang, Qiuchi Li, Lingyu Hua, Dawei Song, Assessing the Memory Ability of Recurrent Neural Networks, Giuseppe De Giacomo, et al. (editors), ECAI 2020: 24th European Conference on Artificial Intelligence,, page 1660,
 * In $$\mathbb R^n$$, a centroid is the mean position of all the points in all of the coordinate directions. The centroid of a subset $$\mathcal X$$ of $$\mathbb R^n$$ is computed as follows:
 * $$\operatorname{Centroid}(\mathcal X)=\frac{\int x g(x)dx}{\int g(x)dx}\quad\quad\quad\quad\quad\quad(6)$$
 * where the integrals are taken over the whole space $$\mathbb R^n$$, and $$g$$ is the characteristic function of the subset, which is 1 inside $$\mathcal X$$ and 0 outside it [27].
 * 1)  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.
 * 2)  the point whose (Cartesian) coordinates are the arithmetic mean of the coordinates of a given finite set of points.
 * 3)  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.
 * 4)  the arithmetic mean (alternatively, median) position of a cluster of points in a coordinate system based on some application-dependent measure of distance.
 * 5)  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.
 * 6) * 1974 [Prentice-Hall],, 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.
 * 1) * 2009, Hao Yuan, Patrick Eugster, An Efficient Algorithm for Solving the Dyck-CFL Reachability Problem on Trees, Giuseppe Castagna (editor), Programming Languages and Systems: 18th European Symposium, Proceedings, Springer, 5502, page 186,
 * A node $$x$$ in a tree $$T$$ is called a centroid of $$T$$ if the removal of $$x$$ will make the size of each remaining connected component no greater than $$\vert T \vert / 2$$. A tree may have at most two centroids, and if there are two then one must be a neighbor of the other [6, 5]. Throughout this paper, we specify the centroid to be the one whose numbering is lexicographically smaller (i.e, we number the nodes from 1 to $$n$$). There exists a linear time algorithm to compute the centroid of a tree due to the work of Goldman [21]. We use $$\operatorname{CT}(T)$$ to denote the centroid of $$T$$ computed by the linear time algorithm.
 * 1)  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.
 * 2) * 1974 [Prentice-Hall],, 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.
 * 1) * 2009, Hao Yuan, Patrick Eugster, An Efficient Algorithm for Solving the Dyck-CFL Reachability Problem on Trees, Giuseppe Castagna (editor), Programming Languages and Systems: 18th European Symposium, Proceedings, Springer, 5502, page 186,
 * A node $$x$$ in a tree $$T$$ is called a centroid of $$T$$ if the removal of $$x$$ will make the size of each remaining connected component no greater than $$\vert T \vert / 2$$. A tree may have at most two centroids, and if there are two then one must be a neighbor of the other [6, 5]. Throughout this paper, we specify the centroid to be the one whose numbering is lexicographically smaller (i.e, we number the nodes from 1 to $$n$$). There exists a linear time algorithm to compute the centroid of a tree due to the work of Goldman [21]. We use $$\operatorname{CT}(T)$$ to denote the centroid of $$T$$ computed by the linear time algorithm.

Usage notes

 * The term centroid is an approximate synonym of and, applied in mathematically abstract situations where the concepts of mass and gravity are not invoked. It may also be called  or.
 * Another near synonym is, which is differently nuanced. It tends to be used in situations where mass is relevant:
 * In geometry, barycentre is a synonym of centroid.
 * In physics, barycentre refers to the centre of gravity of an object that is not (or is not assumed to be) of uniform density.
 * Specifically in astronomy and astrophysics, barycentre refers to the centre of gravity in a system of (usually two, but possibly more) objects that are in orbit around each other. See also
 * In geography, the of a region of the Earth's surface is the centroid of the radial projection of said region onto sea level. (Depending on the size of the region, it may in fact be significantly below sea level.)


 * Any given tree has either one centroid or two. A tree with one centroid is said to be ; one that has two is.

Translations

 * Chinese:
 * Mandarin:
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 * French:
 * German:
 * Ido:
 * Irish: meánlár
 * Italian: centroide
 * Japanese:
 * Korean:, 모양중심
 * Portuguese: centroide
 * Russian:, ,
 * Spanish: centroide
 * Vietnamese: