Centroid

In mathematics and physics, the centroid or geometric center of a plane figure is the arithmetic mean ("average") position of all the points in the shape. The definition extends to any object in n-dimensional space: its centroid is the mean position of all the points in all of the coordinate directions. Informally, it is the point at which an infinitesimally thin cutout of the shape could be perfectly balanced on the tip of a pin (assuming uniform density and a uniform gravitational field).

While in geometry the term "barycenter" is a synonym for "centroid", in astrophysics and astronomy, barycenter is the center of mass of two or more bodies which are orbiting each other. 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 known as the region's geographical center.

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.

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