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Inmathematics andphysics, thecentroid, also known asgeometric center orcenter of figure, of aplane figure orsolid figure is themeanposition of all the points in the figure. The same definition extends to any object in-dimensionalEuclidean space.[1]
Ingeometry, one often assumes uniformmass density, in which case thebarycenter orcenter of mass coincides with the centroid. Informally, it can be understood as the point at which a cutout of the shape (with uniformly distributed mass) could be perfectly balanced on the tip of a pin.[2]
In physics, if variations ingravity are considered, then acenter of gravity can be defined as theweighted mean of all pointsweighted by theirspecific weight.
Ingeography, the centroid of a radial projection of a region of the Earth's surface to sea level is the region'sgeographical center.
The term "centroid" was coined in 1814.[3] It is used as a substitute for the older terms "center of gravity" and "center of mass" when the purely geometrical aspects of that point are to be emphasized. The term is particular to the English language; French, for instance, uses "centre de gravité" on most occasions, and other languages use terms of similar meaning.[citation needed]
The center of gravity, as the name indicates, is a notion that arose in mechanics, most likely in connection with building activities. It is uncertain when the idea first appeared, as the concept likely occurred to many people individually with minor differences. Nonetheless, the center of gravity of figures was studied extensively in Antiquity;Bossut creditsArchimedes (287–212 BCE) with being the first to find the centroid of plane figures, although he never defines it.[4] A treatment of centroids of solids by Archimedes has been lost.[5]
It is unlikely that Archimedes learned the theorem that the medians of a triangle meet in a point—the center of gravity of the triangle—directly fromEuclid, as this proposition is not in theElements. The first explicit statement of this proposition is due toHeron of Alexandria (perhaps the first century CE) and occurs in hisMechanics. It may be added, in passing, that the proposition did not become common in the textbooks on plane geometry until the nineteenth century.[citation needed]
The geometric centroid of aconvex object always lies in the object. A non-convex object might have a centroid that is outside the figure itself. The centroid of aring or abowl, for example, lies in the object's central void.
If the centroid is defined, it is afixed point of all isometries in itssymmetry group. In particular, the geometric centroid of an object lies in the intersection of all itshyperplanes ofsymmetry. The centroid of many figures (regular polygon,regular polyhedron,cylinder,rectangle,rhombus,circle,sphere,ellipse,ellipsoid,superellipse,superellipsoid, etc.) can be determined by this principle alone.
In particular, the centroid of aparallelogram is the meeting point of its twodiagonals. This is not true of otherquadrilaterals.
For the same reason, the centroid of an object withtranslational symmetry is undefined (or lies outside the enclosing space), because a translation has no fixed point.
The centroid of a triangle is the intersection of the threemedians of the triangle (each median connecting a vertex with the midpoint of the opposite side).[6]
For other properties of a triangle's centroid, seebelow.
The centroid of a uniformly denseplanar lamina, such as in figure (a) below, may be determined experimentally by using aplumbline and a pin to find the collocated 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, off the presumed centroid in such a way that it can freely rotate around the pin; the plumb line is then dropped from the pin (figure b). The position of the plumbline is traced on the surface, and the procedure is repeated with the pin inserted at any different point (or a number of points) off the centroid of the object. The unique intersection point of these lines will be the centroid (figure c). Provided that the body is of uniform density, all lines made this way will include the centroid, and all lines will cross at exactly the same place.
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| (a) | (b) | (c) |
This method can be extended (in theory) to concave shapes where the centroid may lie outside the shape, and virtually to solids (again, of uniform density), where the centroid may lie within the body. The (virtual) positions of the plumb lines need to be recorded by means other than by drawing them along the shape.
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 (and exactly at the point where the shape would balance on a pin). In principle, progressively narrower cylinders can be used to find the centroid to arbitrary precision. In practice air currents make this infeasible. However, by marking the overlap range from multiple balances, one can achieve a considerable level of accuracy.
The centroid of a finite set of points in is[1]This point minimizes the sum of squared Euclidean distances between itself and each point in the set.
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 if belongs to and 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 anylist 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 isThe vertical position of the centroid is found in the same way.
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-dimensionalmeasures of the parts.
The centroid of a subset of can also be computed by the vector formula
where theintegrals are taken over the whole space and is thecharacteristic function of the subset of if and otherwise.[7] Note that the denominator is simply themeasure of the set This formula cannot be applied if the set has zero measure, or if either integral diverges.
Alternatively, the coordinate-wise formula for the centroid is defined as
where is theth coordinate of and is the measure of the intersection of with thehyperplane defined by the equation Again, the denominator is simply the measure of
For a plane figure, in particular, the barycentric coordinates are
where is the area of the figure is the length of the intersection of with the vertical line atabscissa and is the length of the intersection of with the horizontal line atordinate
The centroid of a region bounded by the graphs of thecontinuous functions and such that on the interval is given by[7][8]
where is the area of the region (given by).[9][10]
Anintegraph (a relative of theplanimeter) can be used to find the centroid of an object of irregular shape with smooth (or piecewise smooth) boundary. The mathematical principle involved is a special case ofGreen's theorem.[11]
This is a method of determining the centroid of an L-shaped object.
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The centroid of atriangle is the point of intersection of itsmedians (the lines joining eachvertex with the midpoint of the opposite side).[6] The centroid divides each of the medians in theratio which is to say it is located of the distance from each side to the opposite vertex (see figures at right).[12][13] ItsCartesian coordinates are themeans of the coordinates of the three vertices. That is, if the three vertices are and then the centroid (denoted here but most commonly denoted intriangle geometry) is
The centroid is therefore at inbarycentric coordinates.
Intrilinear coordinates the centroid can be expressed in any of these equivalent ways in terms of the side lengths and vertex angles:[14]
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 uniformlinear density, then the center of mass lies at theSpieker center (theincenter of themedial triangle), which does not (in general) coincide with the geometric centroid of the full triangle.
The area of the triangle is times the length of any side times the perpendicular distance from the side to the centroid.[15]
A triangle's centroid lies on itsEuler line between itsorthocenter and itscircumcenter exactly twice as close to the latter as to the former:[16][17]
In addition, for theincenter andnine-point center we have
If is the centroid of the triangle then
Theisogonal conjugate of a triangle's centroid is itssymmedian point.
Any of the three medians through the centroid divides the triangle's area in half. This is not true for other lines through the centroid; the greatest departure from the equal-area division occurs when a line through the centroid is parallel to a side of the triangle, creating a smaller triangle and atrapezoid; in this case the trapezoid's area is that of the original triangle.[18]
Let be any point in the plane of a triangle with vertices and centroid Then the sum of the squared distances of from the three vertices exceeds the sum of the squared distances of the centroid from the vertices by three times the squared distance between and:[19]
The sum of the squares of the triangle's sides equals three times the sum of the squared distances of the centroid from the vertices:[19]
A triangle's centroid is the point that maximizes the product of the directed distances of a point from the triangle's sidelines.[20]
Let be a triangle, let be its centroid, and let be the midpoints of segments respectively. For any point in the plane of[21]
The centroid of a non-self-intersecting closedpolygon defined by vertices is the point[22] where
and
and where is the polygon's signed area,[22] as described by theshoelace formula:
In these formulae, the vertices are assumed to be numbered in order of their occurrence along the polygon's perimeter; furthermore, the vertex is assumed to be the same as meaning on the last case must loop around to (If the points are numbered in clockwise order, the area computed as above, will be negative; however, the centroid coordinates will be correct even in this case.)
The centroid of a non-triangular polygon is not the same as itsvertex centroid, considering only itsvertex set (as thecentroid of a finite set of points;see also:Polygon#Centroid).
The centroid of acone orpyramid is located on the line segment that connects theapex to the centroid of the base. For a solid cone or pyramid, the centroid is 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 the distance from the base plane to the apex.
Atetrahedron is an object inthree-dimensional space having four triangles as itsfaces. A line segment joining a vertex of a tetrahedron with the centroid of the opposite face is called amedian, and a line segment joining the midpoints of two opposite edges is called abimedian. Hence there are four medians and three bimedians. These seven line segments all meet at thecentroid of the tetrahedron.[23] The medians are divided by the centroid in the ratio The centroid of a tetrahedron is the midpoint between itsMonge point and circumcenter (center of the circumscribed sphere). These three points define theEuler line of the tetrahedron that is analogous to theEuler line of a triangle.
These results generalize to any-dimensionalsimplex in the following way. If the set of vertices of a simplex is then considering the vertices asvectors, 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 equal masses.
The centroid of a solid hemisphere (i.e. half of a solid ball) divides the line segment connecting the sphere's center to the hemisphere's pole in the ratio (i.e. it lies of the way from the center to the pole).The centroid of a hollow hemisphere (i.e. half of a hollow sphere) divides the line segment connecting the sphere's center to the hemisphere's pole in half.