ycentroid

The y-centroid, often written as ȳ, is the vertical component of the centroid of a planar region or object. The centroid is the point where the area can be considered balanced, or, for uniform density, the center of mass.

For a region R with area A, the y-centroid is defined by ȳ = (1/A) ∫∫_R y dA. If the region has a nonuniform density described by a function ρ(x,y), the y-coordinate of the center of mass is ȳ = (1/M) ∫∫_R y ρ(x,y) dA, where M = ∫∫_R ρ(x,y) dA is the total mass.

For polygons specified by vertex coordinates (x_i, y_i) with i from 0 to n−1 (and x_n = x_0, y_n = y_0), the signed area is A = (1/2) Σ_i (x_i y_{i+1} − x_{i+1} y_i). The y-coordinate of the centroid is ȳ = (1/(6A)) Σ_i (y_i + y_{i+1}) (x_i y_{i+1} − x_{i+1} y_i). If the vertices are listed in clockwise order and A is negative, use the absolute value of A.

For simple shapes, ȳ has intuitive values: a rectangle of height h with its base on y = 0 has ȳ = h/2; a triangle with base on y = 0 and apex at height h has ȳ = h/3 from the base.

Applications abound in engineering and science, including locating centers of gravity, analyzing bending in beams, and in image analysis where ycentroid refers to the vertical position of a region’s intensity-weighted centroid.