In
geometryGeometry 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 or two-dimensional shape
X is the intersection of all straight lines that divide
X into two parts of equal
momentIn mathematics, a moment is, loosely speaking, a quantitative measure of the shape of a set of points. The "second moment", for example, is widely used and measures the "width" of a set of points in one dimension or in higher dimensions measures the shape of a cloud of points as it could be fit by...
about the line. Informally, it is the "average" (
arithmetic meanIn mathematics and statistics, the arithmetic mean, often referred to as simply the mean or average when the context is clear, is a method to derive the central tendency of a sample space...
) of all points of
X. The definition extends to any object
X in
n-
dimensionIn physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...
al space: its centroid is the intersection of all
hyperplaneA hyperplane is a concept in geometry. It is a generalization of the plane into a different number of dimensions.A hyperplane of an n-dimensional space is a flat subset with dimension n − 1...
s that divide
X into two parts of equal moment.
In
physicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
, the word
centroid means the geometric center of the object's shape, as above, but
barycenter may also mean its physical
center of massIn physics, the center of mass or barycenter of a system is the average location of all of its mass. In the case of a rigid body, the position of the center of mass is fixed in relation to the body...
or the
center of gravityIn physics, a center of gravity of a material body is a point that may be used for a summary description of gravitational interactions. In a uniform gravitational field, the center of mass serves as the center of gravity...
, depending on the context. Informally, the center of mass (and center of gravity in a uniform gravitational field) is the average of all points, weighted by the local
densityThe mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...
or
specific weightThe specific weight is the weight per unit volume of a material. The symbol of specific weight is γ ....
. If a physical object has uniform
densityThe mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...
, then its
center of massIn physics, the center of mass or barycenter of a system is the average location of all of its mass. In the case of a rigid body, the position of the center of mass is fixed in relation to the body...
is the same as the centroid of its shape.
In geography, the centroid of a region of the Earth's surface, projected radially onto said surface, is known as its geographical center.
Properties
The geometric centroid of a
convexIn 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. A non-convex object might have a centroid that is outside the figure itself. The centroid of a
ringIn mathematics, an annulus is a ring-shaped geometric figure, or more generally, a term used to name a ring-shaped object. Or, it is the area between two concentric circles...
or a
bowlA bowl is a common open-top container used in many cultures to serve food, and is also used for drinking and storing other items. They are typically small and shallow, although some, such as punch bowls and salad bowls, are larger and often intended to serve many people.Bowls have existed for...
, for example, lies in the object's central void.
If the centroid is defined, it is a
fixed point of all isometriesA 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 groupThe symmetry group of an object is the group of all isometries under which it is invariant with composition as the operation...
. In particular, the geometric centroid of an object lies in the intersection of all its
hyperplaneA hyperplane is a concept in geometry. It is a generalization of the plane into a different number of dimensions.A hyperplane of an n-dimensional space is a flat subset with dimension n − 1...
s of
symmetrySymmetry 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 polygonA regular polygon is a polygon that is equiangular and equilateral . Regular polygons may be convex or star.-General properties:...
,
regular polyhedronA regular polyhedron is a polyhedron whose faces are congruent 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. it is transitive on its flags...
,
cylinderA cylinder is one of the most basic curvilinear geometric shapes, the surface formed by the points at a fixed distance from a given line segment, the axis of the cylinder. The solid enclosed by this surface and by two planes perpendicular to the axis is also called a cylinder...
,
rectangleIn Euclidean plane geometry, a rectangle is any quadrilateral with four right angles. The term "oblong" is occasionally used to refer to a non-square rectangle...
,
rhombusIn Euclidean geometry, a rhombus or rhomb is a convex quadrilateral whose four sides all have the same length. The rhombus is often called a diamond, after the diamonds suit in playing cards, or a lozenge, though the latter sometimes refers specifically to a rhombus with a 45° angle.Every...
, circle, sphere,
ellipseIn geometry, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis...
,
ellipsoid, superellipse,
superellipsoidIn mathematics, a super-ellipsoid or superellipsoid is a solid whose horizontal sections are super-ellipses 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.
In particular, the centroid of a
parallelogramIn Euclidean geometry, a parallelogram is a convex quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure...
is the meeting point of its two
diagonalA diagonal is a line joining two nonconsecutive vertices of a polygon or polyhedron. Informally, any sloping line is called diagonal. The word "diagonal" derives from the Greek διαγώνιος , from dia- and gonia ; it was used by both Strabo and Euclid to refer to a line connecting two vertices of a...
s. This is not true for other
quadrilateralIn Euclidean plane geometry, a quadrilateral is a polygon with four sides and four vertices or corners. Sometimes, the term quadrangle is used, by analogy with triangle, and sometimes tetragon for consistency with pentagon , hexagon and so on...
s.
For the same reason, the centroid of an object with
translational symmetryIn 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.
Plumb line method
The centroid of a uniform two-dimensional lamina, such as (a) below, may be determined, experimentally, by using a
plumblineA plumbline is a string with a lead weight or plumb-bob, used to provide a vertical reference line.It may also refer to:*The Plumbline, a joke newspaper produced by the McMaster Engineering Society...
and a pin to find the center of mass of a thin body of uniform density having the same shape. The body is held by the pin inserted at a point near the body's perimeter, in such a way that it can freely rotate around the pin; and the plumb line is dropped from the pin (b). The position of the plumbline is traced on the body. The experiment is repeated with the pin inserted at a different point of the object. The intersection of the two lines is the centroid of the figure (c).
This method can be extended (in theory) to concave shapes where the centroid lies outside the shape, and to solids (of uniform density), but the positions of the plumb lines need to be recorded by means other than drawing.
Balancing Method
For convex two-dimensional shapes, the centroid can be found by balancing the shape on a smaller shape, such as the top of a narrow cylinder. The centroid occurs somewhere within the range of contact between the two shapes. In principle, progressively narrower cylinders can be used to find the centroid to arbitrary accuracy. In practice air currents make this unfeasible. However, by marking the overlap range from multiple balances, one can achieve a considerable level of accuracy.
Of a finite set of points
The centroid of a finite set of
points
in
is
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
-
Holes in the figure
, overlaps between the parts, or parts that extend outside the figure can all be handled using negative areas
. Namely, the measures
should be taken with positive and negative signs in such a way that the sum of the signs of
for all parts that enclose a given point
is 1 if
belongs to
, and 0 otherwise.
For example, the figure below (a) is easily divided into a square and a triangle, both with positive area; and a circular hole, with negative area (b).
The centroid of each part can be found in any
list of centroids of simple shapes (c). Then the centroid of the figure is the weighted average of the three points. The horizontal position of the centroid, from the left edge of the figure is
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
measureIn mathematical analysis, 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. In this sense, a measure is a generalization of the concepts of length, area, and volume...
s of the parts.
By integral formula
The centroid of a subset
X of
can also be computed by the
integralIntegration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
where the integral is taken over the whole space
, and
g is the
characteristic function of the subset, which is 1 inside
X and 0 outside it. Note that the denominator is simply the
measureIn mathematical analysis, 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. In this sense, a measure is a generalization of the concepts of length, area, and volume...
of the set
X (However, this formula cannot be applied if the set
X has zero measure, or if either integral diverges.)
Another formula for the centroid is
where
Ck is the
kth coordinate of
C, and
Sk(
z) is the measure of the intersection of
X with the hyperplane defined by the equation
xk =
z. Again, the denominator is simply the measure of
X.
For a plane figure, in particular, the barycenter coordinates are
where
A is the area of the figure
X;
Sy(
x) is the length of the intersection of
X with the vertical line at
abscissaIn mathematics, abscissa refers to that element of an ordered pair which is plotted on the horizontal axis of a two-dimensional Cartesian coordinate system, as opposed to the ordinate...
x; and
Sx(
y) is the analogous quantity for the swapped axes.
Of an L-shaped object
This is a method of determining the centroid of an L-shaped object.
- Divide the shape into two rectangles, as shown in fig 2. Find the centroids of these two rectangles by drawing the diagonals. Draw a line joining the centroids. The centroid of the shape must lie on this line AB.
- Divide the shape into two other rectangles, as shown in fig 3. Find the centroids of these two rectangles by drawing the diagonals. Draw a line joining the centroids. The centroid of the L-shape must lie on this line CD.
- As the centroid of the shape must lie along AB and also along CD, it is obvious that it is at the intersection of these two lines, at O. The point O might not lie inside the L-shaped object.
Of triangle and tetrahedron
The centroid of a
triangleA triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....
is the point of intersection of its
mediansIn 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
vertexIn geometry, a vertex is a special kind of point that describes the corners or intersections of geometric shapes.-Of an angle:...
with the midpoint of the opposite side). The centroid divides each of the medians in the
ratioIn mathematics, a ratio is a relationship between two numbers of the same kind , usually expressed as "a to b" or a:b, sometimes expressed arithmetically as a dimensionless quotient of the two which explicitly indicates how many times the first number contains the second In mathematics, a ratio is...
2:1, which is to say it is located ⅓ of the perpendicular distance between each side and the opposing point (see figures at right). Its Cartesian coordinates are the
meansIn mathematics and statistics, the arithmetic mean, often referred to as simply the mean or average when the context is clear, is a method to derive the central tendency of a sample space...
of the coordinates of the three vertices. That is, if the three vertices are
,
, and
, then the centroid is
The centroid is therefore at
in
barycentric coordinatesIn geometry, the barycentric coordinate system is a coordinate system in which the location of a point is specified as the center of mass, or barycenter, of masses placed at the vertices of a simplex . Barycentric coordinates are a form of homogeneous coordinates...
.
The centroid is also the physical center of mass if the triangle is made from a uniform sheet of material; or if all the mass is concentrated at the three vertices, and evenly divided among them. On the other hand, if the mass is distributed along the triangle's perimeter, with uniform
linear densityLinear density, linear mass density or linear mass is a measure of mass per unit of length, and it is a characteristic of strings or other one-dimensional objects. The SI unit of linear density is the kilogram per metre...
, the center of mass may not coincide with the geometric centroid.
The area of the triangle is 1.5 times the length of any side times the perpendicular distance from the side to the centroid.
Similar results hold for a
tetrahedronIn geometry, a tetrahedron is a polyhedron composed of four triangular 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
n-dimensional
simplexIn geometry, a simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimension. Specifically, an n-simplex is an n-dimensional polytope which is the convex hull of its n + 1 vertices. For example, a 2-simplex is a triangle, a 3-simplex is a tetrahedron,...
in the obvious way. If the set of vertices of a simplex is
, then considering the vertices as vectors, the centroid is
The geometric centroid coincides with the center of mass if the mass is uniformly distributed over the whole simplex, or concentrated at the vertices as
n equal masses.
The
isogonal conjugateIn 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. These three reflected lines concur at the isogonal conjugate of P...
of a triangle's centroid is its
symmedian pointSymmedians are three particular geometrical lines associated with every triangle. They are constructed by taking a median of the triangle , and reflecting the line over the corresponding angle bisector...
.
Centroid of polygon
The centroid of a non-self-intersecting closed polygon defined by
n vertices (
x0,
y0), (
x1,
y1), ..., (
xn−1,
yn−1) is the point (
Cx,
Cy), where
-
-
and where
A is the polygon's signed area,
-
In these formulas, the vertices are assumed to be numbered in order of their occurrence along the polygon's perimeter, and the vertex (
xn ,
yn ) is assumed to be the same as (
x0 ,
y0 ). Note that if the points are numbered in clockwise order the area
A, computed as above, will have a negative sign; but the centroid coordinates will be correct even in this case.
Centroid of cone or pyramid
The centroid of a cone or pyramid is located on the line segment that connects the
apexIn geometry, an apex is the vertex which is in some sense the highest of the figure to which it belongs.*In an isosceles triangle, the apex is the vertex where the two sides of equal length meet, opposite the unequal third side....
to the centroid of the base. For a solid cone or pyramid, the centroid is 1/4 the distance from the base to the apex. For a cone or pyramid that is just a shell (hollow) with no base, the centroid is 1/3 the distance from the base plane to the apex.
See also
- List of centroids
- Fréchet mean
The Fréchet mean , is the point, x, that minimizes the Fréchet function, in cases where such a unique minimizer exists. The value at a point p, of the Fréchet function associated to a random point X on a complete metric space is the expected squared distance from p to X...
- Pappus's centroid theorem
In mathematics, Pappus' centroid theorem is either of two related theorems dealing with the surface areas and volumes of surfaces and solids of revolution....
- K-means algorithm
In statistics and data mining, k-means clustering is a method of cluster analysis which aims to partition n observations into k clusters in which each observation belongs to the cluster with the nearest mean...
- Triangle center
In geometry a triangle center is a point in the plane that is in some sense a center of a triangle akin to the centers of squares and circles. For example the centroid, circumcenter, incenter and orthocenter were familiar to the ancient Greeks, and can be obtained by simple constructions. Each of...
- Chebyshev center
In geometry, the Chebyshev center of a bounded set Q having non-empty interior is the center of the minimal-radius ball enclosing the entire set Q, or, alternatively, the center of largest inscribed ball of Q ....
External links