Barycentriske

Barycentriske coordinates, or barycentric coordinates, are a coordinate system for describing a point relative to a reference triangle (or more generally, a simplex). Any point P in the plane of triangle ABC can be written as P = αA + βB + γC, where α, β, γ are real numbers called the barycentric coordinates. When α + β + γ = 1, the coordinates are normalized and P is an affine combination of the triangle’s vertices. If all three coordinates are nonnegative, P lies inside the triangle; if one is zero, P lies on an edge; negative values place P outside.

A common interpretation is via areas: α = [PBC]/[ABC], β = [PCA]/[ABC], γ = [PAB]/[ABC], where [XYZ] denotes the signed area of triangle XYZ. Equivalently, the normalized coordinates can be found by solving P = αA + βB + γC with α + β + γ = 1. Barycentric coordinates are affine-invariant, meaning they behave well under affine transformations.

Properties and applications: Because they provide linear interpolation of quantities defined at the triangle’s vertices, barycentric coordinates are widely used to interpolate textures, colors, or other attributes across a triangle in computer graphics and in finite element methods. They also offer a convenient description of lines and regions bounded by the triangle, through inequalities on α, β, γ.

Generalization: Barycentric coordinates extend to higher dimensions relative to a simplex. For an n-dimensional simplex with vertices Vi, a point P in its affine hull has weights αi with sum αi = 1, P = Σ αiVi, and interior points satisfy αi ≥ 0. The concept also relates to the center of mass of point masses placed at the vertices.