Barycentric Coordinate System (mathematics) - Barycentric Coordinates On Triangles

Barycentric Coordinates On Triangles

In the context of a triangle, barycentric coordinates are also known as area coordinates, because the coordinates of P with respect to triangle ABC are proportional to the (signed) areas of PBC, PCA and PAB. Areal and trilinear coordinates are used for similar purposes in geometry.

Barycentric or areal coordinates are extremely useful in engineering applications involving triangular subdomains. These make analytic integrals often easier to evaluate, and Gaussian quadrature tables are often presented in terms of area coordinates.

First let us consider a triangle T defined by three vertices, and . Any point located on this triangle may then be written as a weighted sum of these three vertices, i.e.

where, and are the area coordinates (usually denoted as ). These are subjected to the constraint

which means that

Following this, the integral of a function on T is

Note that the above has the form of a linear interpolation. Indeed, area coordinates will also allow us to perform a linear interpolation at all points in the triangle if the values of the function are known at the vertices.