Centroid

created by Webster 1913

(thing) by tongpoo (3.1 mon) (print) ? (I like it!) Tue Dec 11 2001 at 18:44:12

(Mathematics - Affine Geometry)

Centroid of a triangle

The centroid of a triangle is the concurrent point of the three medians, or lines that pass through a vertex and the midpoint of the side opposite of that vertex. The centroid G of a triangle ABC is the point G = (1/3)(A + B + C), which is equal to G = (1/3)(A) + (2/3)(1/2)(B + C), which shows that G lies on the median that passes A and the midpoint A' of B and C, at a point in between A and A' that is closer to A' than it is to A by half the distance. Because of the symmetry of the equation, G lies on all three medians.

Centroid of a finite set of points

To further generalise the definition, for a finite set of points A₁, A₂, … A_n, the centroid is defined as (1/n)(A₁ + A₂ + … A_n). In fact, a midpoint is the centroid of two points. An interesting fact is that the centroid of the roots of a complex polynomial of degree greater than 1 is the same as the centroid of the roots of the derivative function.

Centroid of weighted points

If points A₁, A₂, … A_n had weights w₁, w₂, … w_n, the centroid G of the set of weighted points is: G = (1/w_total)(w₁A₁ + w₂A₂ + … w_nA_n) where w_total = w₁ + w₂ + … w_n.

We can partition the set of weighted points {A₁, … A_n} into two sets, α and β. Suppose that sets A, α, β have total weights A_w, α_w, β_w and centroids A_G, α_G, β_G. Then it turns out that A_G = (1/A_w) ( α_w * α_G + β_w * β_G ) . G lies on the line that passes α_G and β_G. If we consider a triangle centroid to be the centroid of three points with weights 1, then by the above equation it is clear that the centroid lies on the medians at a location that leans closer towards the midpoints.

Other terms for centroid

Another term for the centroid of weighted points is affine combination.

When the weights of an affine combination are nonnegative, then the combination is called a convex combination.

Another term for the centroid of non-weighted points on the Real line is arithmetic mean.

In physics, centroid is also known as center of mass.

(definition) by Webster 1913 (print) Tue Dec 21 1999 at 22:24:51

Cen"troid (?), n. [L. centrum + -oid.]

The center of mass, inertia, or gravity of a body or system of bodies.

printable version
chaos

Circumcenter	measures of central tendency	center of mass	Napoleon's theorem
arithmetic mean	Median	Pappus-Guldinus Theorem	Orthocenter
Theorem of Snapper	affine geometry	Euler line	Pappus' Theorems
convex combination	AM-GM inequality	triangle	Variance
poo	vector	cross ratio	brachistochrone
ham sandwich theorem	moment	affine combination

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