Barycentric coordinates of Poisson–Riemann

The article deals with the problem of finding barycentric coordinates for arbitrary, simply connected, closed, discrete regions that are defined in $\mathbb{R}^2$ and $\mathbb{R}^3$. Barycentric coordinates are given by a set of scalar parameters that unambiguously define a point of the affine space inside a simply connected, closed, discrete region through a specified point basis, which is given by the vertices of the region. Barycentriс coordinates being defined for the simply connected, closed, discrete region are harmonic and satisfy the properties of affine invariance, positive definiteness and equality to unit. The solution is based on the Riemann theorem on the uniqueness of conformal mapping and the Poisson integral formula for the ball. The paper shows the examples of approximation of the potential inside arbitrary, simply connected, closed, discrete regions using the proposed method, compared with the approximation using the finite element method.