Centroid

Centroid of a triangle

In geometry, the centroid or barycenter of an object $X$ in $n$ -dimensional space is the intersection of all hyperplanes that divide $X$ into two parts of equal moment about the hyperplane. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position In mathematics the dimension of a Space is roughly defined as the minimum number of Coordinates needed to specify every point within it Space is the extent within which Matter is physically extended and objects and Events have positions relative to one another 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 Informally, it is the "average" of all points of $X$ . In Mathematics, an average, or central tendency of a Data set refers to a measure of the "middle" or " expected " value of

The centroid of an 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. The density of a material is defined as its Mass per unit Volume: \rho = \frac{m}{V} Different materials usually have different 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 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

In geography, the centroid of a region of the Earth's surface is known as its geographical center.

The centroid of a convex object always lies in the object. 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 A non-convex 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.

1 Centroid of triangle and tetrahedron
- 1.1 Proof that the centroid of a triangle divides each median in the ratio 2:1
2 Centroid of polygon
3 Centroid of a finite set of points
4 Area centroid
5 Integral formula
6 Centroid of cone and pyramid
7 Center of symmetry
8 See also
9 References
10 External links

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). A triangle is one of the basic Shapes of Geometry: a Polygon with three corners or vertices and three sides or edges which are Line In Geometry, a median of a Triangle is a Line segment joining a vertex to the Midpoint of the opposing side In Geometry, a vertex (plural "vertices" is a special kind of point. 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. A ratio is an expression which compares quantities relative to each other (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. In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane 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 That is, if the three vertices are located at $(x a, y a)$ , $(x b, y b)$ , and $(x c, y c)$ , then the centroid is at:

$\Big( \begin{matrix}\frac13\end{matrix} (x_a+x_b+x_c),\; \begin{matrix}\frac13\end{matrix} (y_a+y_b+y_c) \Big) = \begin{matrix}\frac13\end{matrix} (x_a, y_a) + \begin{matrix}\frac13\end{matrix} (x_b, y_b) + \begin{matrix}\frac13\end{matrix} (x_c, y_c).$

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. A tetrahedron (plural tetrahedra) is a Polyhedron composed of four triangular faces three of which meet at each vertex. These line segments are divided by the centroid in the ratio 3:1. The result generalizes to any $n$ -dimensional simplex in the obvious way. In Geometry, a simplex (plural simplexes or simplices) or n -simplex is an n -dimensional analogue of a triangle If the set of vertices of a simplex is $v 0,. . . , v n$ , then considering the vertices as vectors, the centroid is at:

$\frac{1}{n+1}\sum_{i=0}^n v_i$

The isogonal conjugate of a triangle's centroid is its symmedian point. In Geometry, the isogonal conjugate of a point P with respect to a Triangle ABC is constructed by reflecting the lines Symmedians are three particular geometrical lines associated with every triangle.

Proof that the centroid of a triangle divides each median in the ratio 2:1

Let the medians AD, BE and CF of the triangle ABC intersect at G, the centroid of the triangle, and let the straight line AD be extended up to the point O such that

$AG = GO. \,$

Then the triangles AGE and AOC are similar (common angle at A, AO is twice AG, AC is twice AE), and so OC is parallel to GE. Geometry Two geometrical objects are called similar if one is congruent to the result of a uniform scaling (enlarging or shrinking of the other But GE is BG extended, and so OC is parallel to BG. Similarly, OB is parallel to CG.

The figure GBOC is therefore a parallelogram. In Geometry, a parallelogram is a Quadrilateral with two sets of Parallel sides Since the diagonals of a parallelogram bisect one another, the point of intersection D between the diagonals GO and BC is such that GD = DO, and

$GO = GD + DO = 2GD. \,$

So, $AG = GO = 2GD, \,$

or $AG:GD = 2:1. \,$

This is true for every other median. In Geometry, bisection is the division of something into two equal or Congruent parts usually by a line, which is then called a bisector

Centroid of polygon

The centroid of a non-overlapping closed polygon defined by N vertices ( x_i , y_i ) can be calculated as follows. ^[1] The notional vertex ( x_N , y_N ) is the same as ( x₀ , y₀ ).

The area of the polygon is given by:

$A = \frac{1}{2}\sum_{i=0}^{N-1} (x_i\ y_{i+1} - x_{i+1}\ y_i)$

The centroid of the polygon is then given by:

$C_x = \frac{1}{6A}\sum_{i=0}^{N-1}(x_i+x_{i+1})(x_i\ y_{i+1} - x_{i+1}\ y_i)$

$C_y = \frac{1}{6A}\sum_{i=0}^{N-1}(y_i+y_{i+1})(x_i\ y_{i+1} - x_{i+1}\ y_i)$

Centroid of a finite set of points

Given a finite set of points $x_1,x_2,\ldots,x_k$ in $\mathbb{R}^n$ , their centroid $C$ is defined to be

$C = \frac{x_1+x_2+\cdots+x_k}{k}$ .

Area centroid

The centroid of an area is very similar to the center of mass of a body. This is calculated using only the geometry of the figure. If the body is homogeneous, the center of mass will be at the centroid. ^[2]

For a two body figure, you may have an equation that looks like this:

$\overline{y} = \dfrac{\overline{y_1}A_1 + \overline{y_2}A_2}{A_1 + A_2}$

$\overline{y}$ is the distance from your reference coordinate axis to the centroid of the particular area. $A$ is the area of that particular section.

The general function for calculating the centroid of a geometrically complex cross section is most easily applied when the figure is divided into known simple geometries and then applying the formula:

$\overline{x} = \frac{\sum \overline{x_i}A_i}{\sum A_i}$

$\overline{y} = \frac{\sum \overline{y_i}A_i}{\sum A_i}$

The distance from the y-axis to the centroid is $\overline{x}$ . The distance from the x-axis to the centroid is $\overline{y}$ . The coordinates of the centroid are $(\overline{x} , \overline{y})$ .

Integral formula

The abscissa (x coordinate) of the centroid of a plane figure can be given as the integral

$C_x = \frac{\int x f(x) \; dx}{\int f(x) \; dx},$

where f(x) is the extent of the object along the y axis at abscissa x, that is the measure of the figure's section at x. In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane This formula can be derived from the first moment about the y axis of the area. The first moment of area, sometimes misnamed as the first moment of inertia, is based in the mathematical construct moments in metric spaces, stating that the moment

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: $\bar{x}$

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 $\R^n$ , for any dimension n, provided that f(x) is the (n-1)-dimensional measure of the object's cross-section at coordinate x — that is, the set of all points in the object whose first coordinate is x.

Note that the denominator is simply the object's n-dimensional measure. 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 In the special case where f is normalized, i. e. , the denominator is 1, the centroid is called the mean of f. In Statistics, mean has two related meanings the Arithmetic mean (and is distinguished from the Geometric mean or Harmonic mean

The formula cannot be applied if the object has zero measure, or if either integral diverges.

Centroid of cone and pyramid

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.

Center of symmetry

If the centroid is defined, it is a fixed point of all isometries in its symmetry group. A fixed point of an isometry group is a point that is a fixed point for every Isometry in the group The Symmetry group of an object ( Image, signal, etc eg in 1D 2D or 3D is the group of all Isometries under which it is Thus symmetry may fully or partially determine the centroid, depending on the kind of 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 It also follows that for an object with translational symmetry the centroid is undefined, because a translation has no fixed point. In Geometry, a translation "slides" an object by a vector a: T a (p = p + a

References

External links

Encyclopedia of Triangle Centers by Clark Kimberling. The following diagrams depict a list of Centroids. A centroid of an object X in n- Dimensional space is the intersection of Pappus's centroid theorem (also known as the Guldinus theorem, Pappus-Guldinus theorem or Pappus's theorem) is the name of two related Theorems 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
Interactive animations showing Centroid of a triangle and Centroid construction with compass and straightedge

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

Dictionary

-noun

(mathematics, physics) The point at the centre of any shape, sometimes called centre of area or centre of volume. For a triangle, the centroid is the point at which the medians intersect. The co-ordinates of the centroid are the average (arithmetic mean) of the co-ordinates of all the points of the shape. For a shape of uniform density, the centroid coincides with the centre of mass which is also the centre of gravity in a uniform gravitational field.

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