Barycentric coordinates

The points that make up the quadrature rules in this encyclopedia are represented using barycentric coordinates. If ${\mathbf{v}}_0,\dots ,{\mathbf{v}}_{m-1}\in {\mathbb{R}}^d$ are the coordinates of the vertices of a integration domain and $(p_0,\dots ,p_{m-1})\in {\mathbb{R}}^m$ is the barycentric coordinates of a quadrature point, then the coordinates of the corresponding quadrature point on the domain is given by

$$\sum _{i=0}^{m-1}{p}_i{\mathbf{v}}_i.$$

The quadrature weights in this encyclopedia are normalised for a domain of volume 1, so an integral can be approximated by

$${\int }_{I}f(x)\mathrm{d}x\approx v(I)\sum _{i=0}^{n-1}{w}_{i}f({\mathbf{p}}_i),$$

where $\{{\mathbf{p}}_0,\dots ,{\mathbf{p}}_{n-1}\}\subset {\mathbb{R}}^d$ and $\{w_0,\dots ,w_{n-1}\}\subset \mathbb{R}$ are the quadrature points and weights, and $v(I)$ is the volume of the integration domain $I$.