centroide

The centroide is the geometric center of a figure or region, and for a lamina with uniform density it coincides with the center of mass. It is the point where the shape would balance on a sharp pin, and it is also called the centroid or barycenter in many contexts.

For simple shapes, the centroide has straightforward locations. In a triangle, the centroid is the intersection point of the three medians, and it divides each median in a 2:1 ratio from the vertex toward the midpoint of the opposite side. In vector form, if the triangle has vertices A, B, and C, the centroid is G = (A + B + C) / 3. In a rectangle or parallelogram, the centroide lies at the intersection of the diagonals, i.e., at the average of opposite corners.

For a polygon with vertices (x1, y1), (x2, y2), ..., (xn, yn) taken in order, the area A is A = (1/2) sum over i of (xi yi+1 − xi+1 yi), with (xn+1, yn+1) = (x1, y1). The centroid coordinates are

Cx = (1/(6A)) sum over i of (xi + xi+1)(xi yi+1 − xi+1 yi),

Cy = (1/(6A)) sum over i of (yi + yi+1)(xi yi+1 − xi+1 yi).

If the density ρ varies, the center of mass G is found from M = ∫∫ ρ dA, Mx = ∫∫ x ρ dA, My = ∫∫ y ρ dA, with G = (Mx/M, My/M). In three dimensions, the concept extends to solids and polyhedra by volume integrals. Applications appear in physics, engineering, computer graphics, and geometric modeling.