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Centroid

 

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Centroid



 
 
In geometry
Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
, the centroid, geometric center, or barycenter of a plane figure is the intersection of all straight lines that divide into two parts of equal moment
Moment (mathematics)

The concept of moment in mathematics evolved from the concept of moment in physics. The nth moment of a real-valued function f of a real variable about a value c is...
 about the line. Informally, it is the "average
Average

In mathematics, an average, or central tendency of a data set refers to a measure of the "middle" or "Expected value" value of the data set....
" of all points of . The definition extends to any object in -dimension
Dimension

In mathematics, the dimension of a space is roughly defined as the minimum number of coordinates needed to specify every point within it. For example: a point on the unit circle in the plane can be specified by two Cartesian coordinates but one can make do with a single coordinate , so the circle is 1-dimensional even though it exists in...
al space: its centroid is the intersection of all hyperplane
Hyperplane

A hyperplane is a concept in geometry. It is a higher-dimensional generalization of the concepts of a line in the plane and a plane in 3-dimensional space....
s that divide into two parts of equal moment.

In physics
Physics

Physics is the natural science which examines basic concepts such as energy, force, and spacetime and all that derives from these, such as mass, charge, matter and its Motion ....
, the word centroid may mean the geometric center of the object's shape, as above; or its physical barycenter, which is its center of mass
Center of mass

The center of mass of a system of wiktionary:Particles is a specific point at which, for many purposes, the system's mass behaves as if it were concentrated....
 or the center of gravity, depending on the context.






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Triangle
In geometry
Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
, the centroid, geometric center, or barycenter of a plane figure is the intersection of all straight lines that divide into two parts of equal moment
Moment (mathematics)

The concept of moment in mathematics evolved from the concept of moment in physics. The nth moment of a real-valued function f of a real variable about a value c is...
 about the line. Informally, it is the "average
Average

In mathematics, an average, or central tendency of a data set refers to a measure of the "middle" or "Expected value" value of the data set....
" of all points of . The definition extends to any object in -dimension
Dimension

In mathematics, the dimension of a space is roughly defined as the minimum number of coordinates needed to specify every point within it. For example: a point on the unit circle in the plane can be specified by two Cartesian coordinates but one can make do with a single coordinate , so the circle is 1-dimensional even though it exists in...
al space: its centroid is the intersection of all hyperplane
Hyperplane

A hyperplane is a concept in geometry. It is a higher-dimensional generalization of the concepts of a line in the plane and a plane in 3-dimensional space....
s that divide into two parts of equal moment.

In physics
Physics

Physics is the natural science which examines basic concepts such as energy, force, and spacetime and all that derives from these, such as mass, charge, matter and its Motion ....
, the word centroid may mean the geometric center of the object's shape, as above; or its physical barycenter, which is its center of mass
Center of mass

The center of mass of a system of wiktionary:Particles is a specific point at which, for many purposes, the system's mass behaves as if it were concentrated....
 or the center of gravity, depending on the context. Informally, the barycenter is the average of all points, weighted by the local density
Density

The density of a material is defined as its mass per unit volume. The symbol of density is ....
 or specific weight
Specific weight

The specific weight is the weight per unit volume of a material, or:where is the specific weight of the material is the density of the material ...
, respectively.

In geography, the centroid of a region of the Earth's surface, projected radially onto said surface, is known as its geographical center
Geographical centre

In geography, the centroid of a region of the Earth's surface is often known as its geographical centre.*Geographical centre of Europe**Geographical centre of Austria#Center...
.

Properties

The geometric centroid of a convex
Convex set

In Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object....
 object always lies in the object. It is the point of concurrency of the triangle's medians. A non-convex object might have a centroid that is outside the figure itself. The centroid of a ring
Annulus

Annulus , being the Latin and French language for "circle", is a term used to describe various ring or circle shaped objects :* Annulus , the ring-like row of cells surrounding the sorus of ferns and responsible for opening it when ripe...
 or a bowl
Bowl

Bowl may refer to:* Super bowl meaning final game in an NFL season* Bowl, slang meaning "to walk" in the UK: "Let's bowl"* Bowl , a common open-top vessel used to serve food...
, for example, lies in the object's central void.

If the centroid is defined, it is a fixed point of all isometries
Fixed points of isometry groups in Euclidean space

A fixed point of an isometry group is a point that is a Fixed point for every isometry in the group. For any isometry group in Euclidean space the set of fixed points is either empty or an affine space....
 in its symmetry group
Symmetry group

The symmetry group of an object is the group of all isometries under which it is invariant with Function composition as the operation. It is a subgroup of the isometry group of the space concerned....
. In particular, the geometric centroid of an object lies in the intersection of all its hyperplane
Hyperplane

A hyperplane is a concept in geometry. It is a higher-dimensional generalization of the concepts of a line in the plane and a plane in 3-dimensional space....
s of symmetry
Symmetry

Symmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically-pleasing proportionality and balance; such that it reflects beauty or perfection....
. The centroid of many figures (regular polygon
Regular polygon

A regular polygon is a polygon which is Equiangular polygon and equilateral . Regular polygons may be convex or Star polygon....
, regular polyhedron
Regular polyhedron

A regular polyhedron is a polyhedron whose faces are Congruence regular polygons which are assembled in the same way around each vertex. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive - i.e....
, cylinder
Cylinder (geometry)

A cylinder is one of the most curvilinear basic geometric shapes: the surface formed by the points at a fixed distance from a given straight line, the axis of the cylinder....
, rectangle
Rectangle

In geometry, a rectangle is a Closed set planar quadrilateral with four right angles. A rectangle with vertices ABCD would be denoted as .A rectangle with adjacent sides of lengths a and b has area ab and diagonals of equal length ....
, rhombus
Rhombus

In geometry, a rhombus , or rhomb is an equilateral polygon parallelogram. In other words, it is a four-sided polygon in which every side has the same length....
, circle, sphere, ellipse
Ellipse

In mathematics, an ellipse is the apparent shape of a circle viewed obliquely from outside it, as distinct from a hyperbola which is the shape seen from inside....
, ellipsoid
Ellipsoid

An ellipsoid is a type of Quadric that is a higher dimensional analogue of an ellipse. The equation of a standard axis-aligned ellipsoid body in an xyz-Cartesian coordinate system is...
, superellipse, superellipsoid
Superellipsoid

In mathematics, a super-ellipsoid or superellipsoid is solid whose horizontal sections are super ellipse with the same exponent r, and whose vertical sections through the center are super-ellipses with the same exponent t....
, etc.) can be determined by this principle alone.

For the same reason, the centroid of an object with translational symmetry
Translational symmetry

In geometry, a translation "slides" an object by a a: Ta = p + a.In physics and mathematics, continuous translational symmetry is the invariance of a system of equations under any translation....
 is undefined (or lies outside the enclosing space), because a translation has no fixed point.

Centroid of a finite set of points

The centroid of a finite set of points in is

Centroid by geometric decomposition

The centroid of a plane figure can be computed by dividing it into a finite number of simpler figures , computing the centroid and area of each part, and then computing



The same formula holds for any three-dimensional objects, except that each should be the volume of , rather than its area. It also holds for any subset of , for any dimension , with the areas replaced by the -dimensional measure
Measure (mathematics)

In mathematics, more specifically in measure theory, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset....
s of the parts.

This formula holds even if the parts overlap and/or extend outside the set , provided that the measures are taken with positive and negative signs in such a way that the sum of the of all parts that enclose a given point is 1 if belongs to , and 0 otherwise.

Integral formula

The centroid of a subset of can also be computed by the integral
Integral

Integration is an important concept in mathematics, specifically in the field of calculus and, more broadly, mathematical analysis. Given a function ƒ of a Real number variable x and an interval [ab] of the real line, the integral...


where the integral is taken over the whole space , and f is the characteristic function
Characteristic function

In mathematics, characteristic function can refer to any of several distinct concepts:* The most common and universal usage is as a synonym for indicator function, that is the function* The characteristic state function in statistical mechanics....
 of the subset, which is 1 inside and 0 outside it. (However, this formula cannot be applied if the object has zero measure, or if either integral diverges.)

Position of the CM using polar coordinates. (Center of mass).

where denotes the surface mass density (mass/area) of the sheet and is the element of area .

Centroid of triangle and tetrahedron



The centroid of a triangle is the point of intersection of its medians
Median (geometry)

In geometry, a median of a triangle is a line segment joining a vertex to the midpoint of the opposing side. Every triangle has exactly three medians; one running from each vertex to the opposite side....
 (the lines joining each vertex
Vertex (geometry)

In geometry, a vertex is a special kind of point which describes the corners or intersections of geometric shapes. Vertices are commonly used in computer graphics to define the corners of surfaces in 3d models, where each such point is given as a vector....
 with the midpoint of the opposite side). The centroid divides each of the medians in the ratio
Ratio

A ratio is an expression which compares quantities relative to each other. The most common examples involve two quantities, but in theory any number of quantities can be compared....
 2:1, which is to say it is located ? of the perpendicular distance between each side and the opposing point. (As illustrated in the figures to the right).

The centroid is the triangle's center of mass
Center of mass

The center of mass of a system of wiktionary:Particles is a specific point at which, for many purposes, the system's mass behaves as if it were concentrated....
 if the triangle is made from a uniform sheet of material. Its Cartesian coordinates are the means
Arithmetic mean

In mathematics and statistics, the arithmetic mean of a list of numbers is the sum of all of the list divided by the number of items in the list....
 of the coordinates of the three vertices. That is, if the three vertices are , , and , then the centroid is

A similar result holds for a tetrahedron
Tetrahedron

A tetrahedron is a polyhedron composed of four triangle faces, three of which meet at each vertex . A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids....
: its centroid is the intersection of all line segments that connect each vertex to the centroid of the opposite face. These line segments are divided by the centroid in the ratio 3:1. The result generalizes to any -dimensional simplex
Simplex

In geometry, a simplex or n-simplex is an n-dimensional analogue of a triangle. Specifically, a simplex is the convex hull of a set of affine transformation Point s in some Euclidean space of dimension n or higher ....
 in the obvious way. If the set of vertices of a simplex is , then considering the vertices as vectors, the centroid is

The isogonal conjugate
Isogonal conjugate

In geometry, the isogonal conjugate of a point P with respect to a triangle ABC is constructed by reflection the lines PA, PB, and PC about the angle bisectors of A, B, and C....
 of a triangle's centroid is its symmedian point.

Centroid of polygon

The centroid of a non-overlapping closed polygon defined by n vertices ( xi , yi ) can be calculated as follows.. The area of the polygon is



and its centroid is where





In these formulas, the vertex ( xn , yn ) is assumed to be the same as ( x0 , y0 ).

Centroid of cone and pyramid

The centroid of a cone or pyramid is located on the line segment that connects the apex
Apex

Apex may refer to:...
 to the centroid of the base, and divides that segment in the ratio 3:1.

See also

  • List of centroids
    List of centroids

    The following diagrams depict a list of centroids. A centroid of an object in -dimensional space is the intersection of all hyperplanes that divide into two parts of equal moment about the hyperplane....
  • Pappus's centroid theorem
    Pappus's centroid theorem

    Pappus's centroid theorem is the name of two related theorems dealing with the surface areas and volumes of surfaces and solids of revolution....
  • K-means algorithm
    K-means algorithm

    The -means algorithm is an algorithm to data clustering objects based on attributes into partition of a set, . It is similar to the expectation-maximization algorithm for mixtures of Gaussian distribution in that they both attempt to find the centers of natural clusters in the data....


External links

  • by Clark Kimberling. The centroid is indexed as X(2).
  • by Antonio Gutierrez from Geometry Step by Step from the Land of the Incas.
  • at cut-the-knot
    Cut-the-knot

    Cut-the-knot is an educational website maintained by Alexander Bogomolny and devoted to popular exposition of a great variety of topics in mathematics....
  • at cut-the-knot
    Cut-the-knot

    Cut-the-knot is an educational website maintained by Alexander Bogomolny and devoted to popular exposition of a great variety of topics in mathematics....
  • Interactive animations showing and