Information about Centroid
In geometry, the centroid or barycenter of an object 
in 
-dimensional space is the intersection of all hyperplanes that divide into two parts of equal moment about the hyperplane. Informally, it is the "average" of all points of .
The geometric centroid of a physical object coincides with its center of mass if the object has uniform density, or if the object's shape and density have a symmetry which fully determines the centroid. These conditions are sufficient but not necessary.
The centroid of a finite set of points can be computed as the arithmetic mean of each coordinate of the points.
In geography, the centroid of a region of the Earth's surface is known as its geographical centre.
The centroid of a triangle is the point of intersection of its medians (the lines joining each vertex with the midpoint of the opposite side). The centroid divides each of the medians in the ratio 2:1, which is to say it is located ⅓ 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 if the triangle is made from a uniform sheet of material. Its Cartesian coordinates are the means of the coordinates of the three vertices. That is, if the three vertices are located at
, 
, and 
, then the centroid is at
A similar result holds for a tetrahedron: 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 in the obvious way. If the set of vertices of a simplex is 
, then considering the vertices as vectors, the centroid is at
The isogonal conjugate of a triangle's centroid is its symmedian point.
,
and 
of the 
intersect at the point 
.
is the centroid of the 
.
Let the straight line be extended up to the point 
such that 
=
.
Then the figure
will be a parallelogram, its opposite sides being parallel.
Thus, its diagonals and 
bisect one another at the point 
.
Therefore,
=
.
But = = + .
So, = 
Or, 
This is true for every other median.
A centroid is simply where the 3 medians of a triangle intersect.
,
where 
is the vertical extent of the object at abscissa 
. This formula can be derived from the first moment about the y axis of the area.
This process is equivalent to taking a weighted average. Supposing that the y axis represents frequency, and the x axis represents the variable whose average we want to find, then the location of the centroid along the x axis is simply the mean:
Hence the centroid can be thought of as a weighted average of many infintesimally small elements that represent a particular shape.
The same formula yields the first coordinate of the centroid of an object in
, for any dimension , provided that is the 
-dimensional measure of the object's cross-section at coordinate — that is, the set of all points in the object whose first coordinate is .
Note that the denominator is simply the object's-dimensional measure. In the special case where f is normalized, i.e., the denominator is 1, the centroid is called the mean of f.
The formula cannot be applied if the object has zero measure, or if either integral diverges.
A hyperplane is a concept in geometry. It is a higher-dimensional generalization of the concepts of a line in Euclidean plane geometry and a plane in 3-dimensional Euclidean geometry.
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A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex.
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in -dimensional space is the intersection of all hyperplanes that divide into two parts of equal moment about the hyperplane. Informally, it is the "average" of all points of .
The geometric centroid of a physical object coincides with its center of mass if the object has uniform density, or if the object's shape and density have a symmetry which fully determines the centroid. These conditions are sufficient but not necessary.
The centroid of a finite set of points can be computed as the arithmetic mean of each coordinate of the points.
In geography, the centroid of a region of the Earth's surface is known as its geographical centre.
Centroid of triangle and tetrahedron
) ) |
The centroid of a triangle is the point of intersection of its medians (the lines joining each vertex with the midpoint of the opposite side). The centroid divides each of the medians in the ratio 2:1, which is to say it is located ⅓ 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 if the triangle is made from a uniform sheet of material. Its Cartesian coordinates are the means of the coordinates of the three vertices. That is, if the three vertices are located at
, , and , then the centroid is at
A similar result holds for a tetrahedron: 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 in the obvious way. If the set of vertices of a simplex is , then considering the vertices as vectors, the centroid is at
The isogonal conjugate of a triangle's centroid is its symmedian point.
Proof that the centroid of a triangle divides each median in the ratio 2:1
Let the medians, and of the intersect at the point .
is the centroid of the .
Let the straight line
be extended up to the point
such that =.
Then the figure
will be a parallelogram, its opposite sides being parallel.
Thus, its diagonals
and bisect one another at the point .
Therefore,
=.
But
= = + .
So,
=
Or,
This is true for every other median.
Centroids of cones and pyramids
The centroid of a cone or pyramid is located on the line segment that connects the apex to the centroid of the base, and divides that segment in the ratio 3:1.A centroid is simply where the 3 medians of a triangle intersect.
Centroid and convexity
The centroid of a convex object always lies in the object. A concave object might have a centroid that is outside the figure itself. The centroid of a ring or a bowl, for example, lies in the object's central void.Integral formula
The abscissa (x coordinate) of the centroid of a plane figure can be given as the integral,
where is the vertical extent of the object at abscissa . This formula can be derived from the first moment about the y axis of the area.
This process is equivalent to taking a weighted average. Supposing that the y axis represents frequency, and the x axis represents the variable whose average we want to find, then the location of the centroid along the x axis is simply the mean:
Hence the centroid can be thought of as a weighted average of many infintesimally small elements that represent a particular shape.
The same formula yields the first coordinate of the centroid of an object in
, for any dimension , provided that is the -dimensional measure of the object's cross-section at coordinate — that is, the set of all points in the object whose first coordinate is .
Note that the denominator is simply the object's
-dimensional measure. In the special case where f is normalized, i.e., the denominator is 1, the centroid is called the mean of f.
The formula cannot be applied if the object has zero measure, or if either integral diverges.
Center of symmetry
If the centroid is defined, it is a fixed point of all isometries in its symmetry group. Thus symmetry may fully or partially determine the centroid, depending on the kind of symmetry. It also follows that for an object with translational symmetry the centroid is undefined, because a translation has no fixed point.See also
External links
- Encyclopedia of Triangles Centers by Clark Kimberling. The centroid is indexed as X(2).
- Triangle centers by Antonio Gutierrez from Geometry Step by Step from the Land of the Incas.
- Characteristic Property of Centroid at cut-the-knot
- Barycentric Coordinates at cut-the-knot
- Centroid of a triangle With interactive animation
Geometry (Greek γεωμετρία; geo = earth, metria = measure) is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. Geometry is one of the oldest sciences.
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dimension (Latin, "measured out") is a parameter or measurement required to define the characteristics of an object—i.e., length, width, and height or size and shape.
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The term SPACE (capitalized) can refer to:
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- , a Canadian science-fiction channel
- The Society for Promotion of Alternative Computing and Employment
- DSPACE, a term in computational complexity theory
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Not to be confused with Hypersonic aircraft.
A hyperplane is a concept in geometry. It is a higher-dimensional generalization of the concepts of a line in Euclidean plane geometry and a plane in 3-dimensional Euclidean geometry.
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- Principle of Moments redirects here. For the Robert Plant album, see The Principle of Moments. See also Moment (mathematics) for a more abstract concept of moments that evolved from this concept of physics.
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In mathematics, an average, or central tendency of a data set refers to a measure of the "middle" or "expected" value of the data set. There are many different descriptive statistics that can be chosen as a measurement of the central tendency of the data items.
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Physics is the science of matter[1] and its motion[2][3], as well as space and time[4][5] —the science that deals with concepts such as force, energy, mass, and charge.
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center of mass of a system of particles is a specific point at which, for many purposes, the system's mass behaves as if it were concentrated. The center of mass is a function only of the positions and masses of the particles that comprise the system.
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In physics, density is mass m per unit volume V—how heavy something is compared to its size. A small, heavy object, such as a rock or a lump of lead, is denser than a lighter object of the same size or a larger object of the same weight, such as pieces of
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In mathematics and statistics, the arithmetic mean (or simply the mean) of a list of numbers is the sum of all the members of the list divided by the number of items in the list. The arithmetic mean is what students are taught very early to call the "average".
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In geography, the centroid of a region of the Earth's surface is often known as its geographical centre.
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- Geographical centre of Europe
- Centre of Norway
- Centre of Scotland
- Geographical centre of Switzerland
- Centre points of the United Kingdom
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A triangle is one of the basic shapes of geometry: a polygon with three corners or and three sides or edges which are straight line segments.
In Euclidean geometry any three non-collinear points determine a triangle and a unique plane, i.e.
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In Euclidean geometry any three non-collinear points determine a triangle and a unique plane, i.e.
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median of a triangle is a line joining a vertex to the midpoint of the opposite side. It divides the triangle into two parts of equal area. The three medians intersect in the triangle's centroid or center of mass, and two-thirds of the length of each median is between the vertex
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- For other uses of the word, see Vertex.
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center of mass of a system of particles is a specific point at which, for many purposes, the system's mass behaves as if it were concentrated. The center of mass is a function only of the positions and masses of the particles that comprise the system.
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Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane through two numbers, usually called the x-coordinate and the y-coordinate of the point.
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In mathematics and statistics, the arithmetic mean (or simply the mean) of a list of numbers is the sum of all the members of the list divided by the number of items in the list. The arithmetic mean is what students are taught very early to call the "average".
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For the academic journal, see Tetrahedron (journal).
A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex.
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In geometry, a simplex (plural simplexes or simplices) or n-simplex is an n-dimensional analogue of a triangle. Specifically, a simplex is the convex hull of a set of (n
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spatial vector, or simply vector, is a concept characterized by a magnitude and a direction. A vector can be thought of as an arrow in Euclidean space, drawn from an initial point A pointing to a terminal point B.
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In geometry, the isogonal conjugate of a point P with respect to a triangle ABC is constructed by reflecting the lines PA, PB, and PC about the angle bisectors of A, B, and C.
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symmedians. To construct one, start with a median of the triangle (a line connecting one vertex with the midpoint of the opposite side) then reflect it over the corresponding angle bisector (the line through the same vertex that divides the angle of the triangle there in two equal
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Apex may refer to:
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- Apex (mollusc), the tip of the spire of the shell of a gastropod
- Apex (headdress), a pointed piece of olive-wood, the base of which was surrounded with a lock of wool, worn by Roman priests
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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. For example, a solid cube is convex, but anything that is hollow or has a dent in it, for example, a crescent shape, is not convex.
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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. For example, a solid cube is convex, but anything that is hollow or has a dent in it, for example, a crescent shape, is not convex.
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Annulus (or anulus), being the Latin for "ring", is a term used to describe various ring-shaped objects:
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- Annulus (entomology), an antennal unit in simple antennae, or a ring-like marking or structure surrounding a joint or segment
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Bowl may refer to:
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- Bowl (vessel), a common open-top vessel used to serve food
- Bowl (drug culture), the receptacle in which marijuana is placed prior to smoking
- Bowls, a precision sport popular in the United Kingdom
- The Bowl (Utah vs.
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Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane through two numbers, usually called the x-coordinate and the y-coordinate of the point.
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In mathematics the concept of a measure generalizes notions such as "length", "area", and "volume" (but not all of its applications have to do with physical sizes). Informally, given some base set, a "measure" is any consistent assignment of "sizes" to (some of) the subsets of the
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