Editing Barycentric coordinate system (section)

==Barycentric coordinates on tetrahedra==

Barycentric coordinates may be easily extended to [[coordinate space|three dimensions]].  The 3D [[simplex]] is a [[tetrahedron]], a [[polyhedron]] having four triangular faces and four vertices.  Once again, the four barycentric coordinates are defined so that the first vertex <math>\mathbf{r}_1</math> maps to barycentric coordinates <math>\lambda = (1,0,0,0)</math>, <math>\mathbf{r}_2 \to (0,1,0,0)</math>, etc.

This is again a linear transformation, and we may extend the above procedure for triangles to find the barycentric coordinates of a point <math>\mathbf{r}</math> with respect to a tetrahedron:

<math display=block>
\left(\begin{matrix}\lambda_1 \\ \lambda_2 \\ \lambda_3\end{matrix}\right) = \mathbf{T}^{-1} ( \mathbf{r}-\mathbf{r}_4 )
</math>

where <math>\mathbf{T}</math> is now a 3×3 matrix:

<math display=block>
\mathbf{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)
</math>

and <math>\lambda_4 = 1 - \lambda_1 - \lambda_2 - \lambda_3</math>with the corresponding Cartesian coordinates:<math display="block">\begin{align}
x &= \lambda_1 x_1 + \lambda_2 x_2 + \lambda_3 x_3 + (1-\lambda_1-\lambda_2-\lambda_3)x_4 \\
y &= \lambda_1 y_1 + \,\lambda_2 y_2 + \lambda_3 y_3 + (1-\lambda_1-\lambda_2-\lambda_3)y_4 \\
z &= \lambda_1 z_1 + \,\lambda_2 z_2 + \lambda_3 z_3 + (1-\lambda_1-\lambda_2-\lambda_3)z_4
\end{align}</math>Once again, the problem of finding the barycentric coordinates has been reduced to [[matrix inverse|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.