centroid

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centroid

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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.

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centroid (plural centroids)

1. (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 151–152:

     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”, in Zvi Drezner, Horst W. Hamacher, editors, Facility Location: Applications and Theory, Springer-Verlag, 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 CD-ROM 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 ${\displaystyle \mathbb {R} ^{n}}$, a centroid is the mean position of all the points in all of the coordinate directions. The centroid of a subset ${\displaystyle {\mathcal {X}}}$ of ${\displaystyle \mathbb {R} ^{n}}$ is computed as follows:

     ${\displaystyle \operatorname {Centroid} ({\mathcal {X}})={\frac {\int xg(x)dx}{\int g(x)dx}}\quad \quad \quad \quad \quad \quad (6)}$

     where the integrals are taken over the whole space ${\displaystyle \mathbb {R} ^{n}}$, and ${\displaystyle g}$ is the characteristic function of the subset, which is 1 inside ${\displaystyle {\mathcal {X}}}$ and 0 outside it .

2. (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.
3. (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.
4. (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 ${\displaystyle f}$ is given by ${\displaystyle {\frac {\int xf(x)dx}{\int f(x)dx}}}$, where the integrals are calculated over the domain of ${\displaystyle f}$.

5. (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 application-dependent measure of distance.
   • 2011, Ross Maciejewski, Data Representations, Transformations, and Statistics for Visual Reasoning, Morgan & Claypool Publishers, page 34:

     The k-means 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 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”, in Ron Bekkerman, Mikhail Bilenko, John Langford, editors, Scaling Up Machine Learning, Cambridge University Press, page 26:

     The k-means 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 k-means, 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 k-means has two steps. In the cluster assignment step, k-means 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 re-estimation step, k-means re-estimates the new cluster centroids based on the reassignments of points to clusters in the previous step. The cluster re-assignment and centroid re-estimation 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.

6. (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 , 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.

   • 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, LNCS 5502, page 186,

     A node ${\displaystyle x}$ in a tree ${\displaystyle T}$ is called a centroid of ${\displaystyle T}$ if the removal of ${\displaystyle x}$ will make the size of each remaining connected component no greater than ${\displaystyle \vert T\vert /2}$. A tree may have at most two centroids, and if there are two then one must be a neighbor of the other . 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 ${\displaystyle n}$). There exists a linear time algorithm to compute the centroid of a tree due to the work of Goldman . We use ${\displaystyle \operatorname {CT} (T)}$ to denote the centroid of ${\displaystyle T}$ computed by the linear time algorithm.

(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 Two-body 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.

• (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

• barycentre / barycenter
• geographical centre

• 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
• k-means 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

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