centroid
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English[edit]
Etymology[edit]
From centre + oid. From 1844,^{[1]} 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[edit]
centroid (plural centroids)
 (geometry, physics, engineering, of an object or a geometrical figure) 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.
 1892, Leander Miller Hoskins, The Elements of Graphic Statics, MacMillan and Co., pages 151152,
 The center of gravity of any body or geometrical magnitude is by definition the same as the centroid of a certain system of parallel forces. It will be convenient, therefore, to use the word centroid in most cases instead of center of gravity. […] The centroid of any area may be found by the following method: Divide the area into parts such that the area and centroid of each part are known. Take the centroids of the partial areas as the points of application of forces proportional respectively to those areas. The centroid of this system of forces is the centroid of the total area, and may be found by the method of Art. 172.
 2004, Richard L. Francis, Timothy J. Lowe, Arie Tamir, 7: Demand Point Aggregation for Local Models, Zvi Drezner, Horst W. Hamacher (editors), Facility Location: Applications and Theory, SpringerVerlag, page 207,
 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 CDROM phone books (Francis, Lowe, Rushton and Rayco 1999).
 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, IOS Press, page 1660,
 In , a centroid is the mean position of all the points in all of the coordinate directions. The centroid of a subset of is computed as follows:
 where the integrals are taken over the whole space , and is the characteristic function of the subset, which is 1 inside and 0 outside it [27].
 In , a centroid is the mean position of all the points in all of the coordinate directions. The centroid of a subset of is computed as follows:
 1892, Leander Miller Hoskins, The Elements of Graphic Statics, MacMillan and Co., pages 151152,
 (geometry, specifically, of a triangle) 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.
 (of a finite set of points) the point whose (Cartesian) coordinates are the arithmetic mean of the coordinates of a given finite set of points.
 (mathematical analysis, of a function) 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 is given by , where the integrals are calculated over the domain of .
 (statistics, cluster analysis, of a cluster of points) the arithmetic mean (alternatively, median) position of a cluster of points in a coordinate system based on some applicationdependent measure of distance.
 2011, Ross Maciejewski, Data Representations, Transformations, and Statistics for Visual Reasoning, Morgan & Claypool Publishers, page 34,
 The kmeans procedure classifies a given data set by using a user defined number of clusters, k, a priori. The centroids can be placed randomly, or algorithmically, but it should be noted that the initial placement will affect the result. The next step is to analyze each point within the data set and group it with the nearest centroid according to some distance metric. When all points have been assigned to a group, a new centroid is calculated for each group as a barycenter of the cluster, resulting from the previous step. Once the k new centroids are calculated, the algorithm reiterates through the data set, and each sample is again assigned to a cluster based on its distance to the new centroids. This process is continued until the position^{[sic]} of the centroids no longer change.
 2012, Biswanath Panda, Joshua S. Herbach, Sugato Basu, Roberto J. Bayardo, 2: MapReduce and its Application to Massively Parallel Learning of Decision Tree Ensembles, Ron Bekkerman, Mikhail Bilenko, John Langford (editors), Scaling Up Machine Learning, Cambridge University Press, page 26,
 The kmeans clustering algorithm (MacQueen, 1967) is a widely used clustering method that applies relocation of points to find a locally optimal partitioning of a dataset. In kmeans, the total distance between each data point and a representative point (centroid) of the cluster to which it is assigned is minimized. Each iteration of kmeans has two steps. In the cluster assignment step, kmeans assigns each point to a cluster such that, of all the current cluster centroids, the point is closest to the centroid of that cluster. In the cluster reestimation step, kmeans reestimates the new cluster centroids based on the reassignments of points to clusters in the previous step. The cluster reassignment and centroid reestimation steps proceed in iterations until a specified convergence criterion is reached, such as when the total distance between clusters and centroids does not change substantially from one iteration to another.
 2011, Ross Maciejewski, Data Representations, Transformations, and Statistics for Visual Reasoning, Morgan & Claypool Publishers, page 34,
 (graph theory, of a tree) 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 [PrenticeHall], 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 34), 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.
 2009, Hao Yuan, Patrick Eugster, An Efficient Algorithm for Solving the DyckCFL Reachability Problem on Trees, Giuseppe Castagna (editor), Programming Languages and Systems: 18th European Symposium, Proceedings, Springer, LNCS 5502, page 186,
 A node in a tree is called a centroid of if the removal of will make the size of each remaining connected component no greater than . 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 ). There exists a linear time algorithm to compute the centroid of a tree due to the work of Goldman [21]. We use to denote the centroid of computed by the linear time algorithm.
 1974 [PrenticeHall], Narsingh Deo, Graph Theory with Applications to Engineering and Computer Science, 2017, Dover, page 248,
Usage notes[edit]
(centre of gravity and related senses):
 The term centroid is an approximate synonym of centre of gravity and centre of mass, applied in mathematically abstract situations where the concepts of mass and gravity are not invoked. It may also be called centre of area or centre of volume.
 Another near synonym is barycentre, 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 Twobody problem on Wikipedia.Wikipedia
 In geography, the geographical centre 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.)
(graph theory: type of node in a tree):
 Any given tree has either one centroid or two. A tree with one centroid is said to be centroidal; one that has two is bicentroidal.
Synonyms[edit]
 (centre of gravity of a uniform, rigid object): barycentre, centre of area, centre of gravity, centre of mass, centre of volume, geometric centre, geometric centroid
 (centre of gravity of a set of points of equal mass): barycentre, centre of gravity, centre of mass
 (centroid of a function): function centroid
 (cluster theory: arithmetic mean or median position of points in a cluster):
 (graph theory: type of node in a tree): tree centroid
Derived terms[edit]
Translations[edit]
mean position of points in a geometrical figure — See also translations at center of gravity, barycenter

cluster analysis: arithmetic mean or median of points in a cluster
graph theory: type of node in a tree
See also[edit]
References[edit]
Further reading[edit]
 Center of mass on Wikipedia.Wikipedia
 List of centroids on Wikipedia.Wikipedia
 Barycenter on Wikipedia.Wikipedia
 Euler line on Wikipedia.Wikipedia
 Triangle center on Wikipedia.Wikipedia
 Locating the center of mass on Wikipedia.Wikipedia
 kmeans clustering on Wikipedia.Wikipedia
 Barycentric coordinate system on Wikipedia.Wikipedia
 Cluster analysis on Wikipedia.Wikipedia
 Fréchet mean on Wikipedia.Wikipedia
 Centroid on Encyclopedia of Mathematics
 Geometric Centroid on Wolfram MathWorld
 Centroid Point on Wolfram MathWorld
 Triangle Centroid on Wolfram MathWorld
 Function Centroid on Wolfram MathWorld