Centroid

In mathematics and physics, the centroid or geometric center of a plane figure is the arithmetic mean position of all the points in the figure. Informally, it is the point at which a cutout of the shape (with uniformly distributed mass) could be perfectly balanced on the tip of a pin.[1] The same definition extends to any object in n-dimensional space.[2]

While in geometry, the word barycenter is a synonym for centroid, in astrophysics and astronomy, the barycenter is the center of mass of two or more bodies that orbit one another. In physics, the center of mass is the arithmetic mean of all points weighted by the local density or specific weight. If a physical object has uniform density, then its center of mass is the same as the centroid of its shape.

In geography, the centroid of a radial projection of a region of the Earth's surface to sea level is the region's geographical center.

The term "centroid" is of recent coinage (1814).[citation needed] 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 peculiar to the English language. The French use "centre de gravité" on most occasions, and others use terms of similar meaning.

The center of gravity, as the name indicates, is a notion that arose in mechanics, most likely in connection with building activities. When, where, and by whom it was invented is not known, as it is a concept that likely occurred to many people individually with minor differences.

While Archimedes does not state that proposition explicitly, he makes indirect references to it, suggesting he was familiar with it. However, Jean-Étienne Montucla (1725–1799), the author of the first history of mathematics (1758), declares categorically (vol. I, p. 463) that the center of gravity of solids is a subject Archimedes did not touch.

In 1802 Charles Bossut (1730–1813) published a two-volume Essai sur l'histoire générale des mathématiques. This book was highly esteemed by his contemporaries, judging from the fact that within two years after its publication it was already available in translation in Italian (1802–03), English (1803), and German (1804). Bossut credits Archimedes with having found the centroid of plane figures, but has nothing to say about solids.[3]

While it is possible Euclid was still active in Alexandria during the childhood of Archimedes (287–212 BCE), it is certain that when Archimedes visited Alexandria, Euclid was no longer there. Thus Archimedes could not have learned the theorem that the medians of a triangle meet in a point—the center of gravity of the triangle—directly from Euclid, as this proposition is not in Euclid's Elements. The first explicit statement of this proposition is due to Heron of Alexandria (perhaps the first century CE) and occurs in his Mechanics. It may be added, in passing, that the proposition did not become common in the textbooks on plane geometry until the nineteenth century.

The geometric centroid of a convex object always lies in the object. 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.

If the centroid is defined, it is a fixed point of all isometries in its symmetry group. In particular, the geometric centroid of an object lies in the intersection of all its hyperplanes of symmetry. 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 a parallelogram is the meeting point of its two diagonals. This is not true of other quadrilaterals.

For the same reason, the centroid of an object with translational 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 three medians of the triangle (each median connecting a vertex with the midpoint of the opposite side).[4]

The centroid of a uniformly dense planar lamina, such as in figure (a) below, may be determined experimentally by using a plumbline 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.

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.

This point minimizes the sum of squared Euclidean distances between itself and each point in the set.

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

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.

where A is the area of the figure X; Sy(x) is the length of the intersection of X with the vertical line at abscissa x; and Sx(y) is the analogous quantity for the swapped axes.

In trilinear coordinates the centroid can be expressed in any of these equivalent ways in terms of the side lengths a, b, c and vertex angles L, M, N:[11]

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 density, then the center of mass lies at the Spieker center (the incenter of the medial triangle), which does not (in general) coincide with the geometric centroid of the full triangle.

The area of the triangle is 1.5 times the length of any side times the perpendicular distance from the side to the centroid.[12]

A triangle's centroid lies on its Euler line between its orthocenter H and its circumcenter O, exactly twice as close to the latter as to the former:

The isogonal conjugate of a triangle's centroid is its symmedian 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 a trapezoid; in this case the trapezoid's area is 5/9 that of the original triangle.[15]

Let P be any point in the plane of a triangle with vertices A, B, and C and centroid G. Then the sum of the squared distances of P from the three vertices exceeds the sum of the squared distances of the centroid G from the vertices by three times the squared distance between P and G:

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:

A triangle's centroid is the point that maximizes the product of the directed distances of a point from the triangle's sidelines.[17]

Let ABC be a triangle, let G be its centroid, and let D, E, and F be the midpoints of BC, CA, and AB, respectively. For any point P in the plane of ABC then

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),[19] where

and where A is the polygon's signed area,[19] as described by the shoelace formula:

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

A tetrahedron is an object in three-dimensional space having four triangles as its faces. A line segment joining a vertex of a tetrahedron with the centroid of the opposite face is called a median, and a line segment joining the midpoints of two opposite edges is called a bimedian. Hence there are four medians and three bimedians. These seven line segments all meet at the centroid of the tetrahedron.[20] The medians are divided by the centroid in the ratio 3:1. The centroid of a tetrahedron is the midpoint between its Monge point and circumcenter (center of the circumscribed sphere). These three points define the Euler line of the tetrahedron that is analogous to the Euler line of a triangle.

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+1 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 3:5 (i.e. it lies 3/8 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.