Barycentric Coordinate System (mathematics) - Barycentric Coordinates On Tetrahedra

Barycentric Coordinates On Tetrahedra

Barycentric coordinates may be easily extended to three dimensions. The 3D simplex is a tetrahedron, a polyhedron having four triangular faces and four vertices. Once again, the barycentric coordinates are defined so that the first vertex maps to barycentric coordinates, etc.

This is again a linear transformation, and we may extend the above procedure for triangles to find the barycentric coordinates of a point with respect to a tetrahedron:

$\left(\begin{matrix}\lambda_1 \\ \lambda_2 \\ \lambda_3\end{matrix}\right) = \textbf{T}^{-1} ( \textbf{r}-\textbf{r}_4 ) \,$

where is now a 3×3 matrix:

$\textbf{T} = \left(\begin{matrix} x_1-x_4 & x_2-x_4 & x_3-x_4\\ y_1-y_4 & y_2-y_4 & y_3-y_4\\ z_1-z_4 & z_2-z_4 & z_3-z_4 \end{matrix}\right)$

Once again, the problem of finding the barycentric coordinates has been reduced to inverting a 3×3 matrix. 3D barycentric coordinates may be used to decide if a point lies inside a tetrahedral volume, and to interpolate a function within a tetrahedral mesh, in an analogous manner to the 2D procedure. Tetrahedral meshes are often used in finite element analysis because the use of barycentric coordinates can greatly simplify 3D interpolation.

Read more about this topic: Barycentric Coordinate System (mathematics)