Editing Barycentric coordinate system (section)

====Interpolation on a triangular unstructured grid ====
[[File:Piecewise linear function2D.svg|thumb|upright=1.3|Surface (upper part) obtained from linear interpolation over a given triangular grid (lower part) in the ''x'',''y'' plane. The surface approximates a function ''z''=''f''(''x'',''y''), given only the values of ''f'' on the grid's vertices.]]
If <math>f(\mathbf{r}_1),f(\mathbf{r}_2),f(\mathbf{r}_3)</math> are known quantities, but the values of {{mvar|f}} inside the triangle defined by <math>\mathbf{r}_1,\mathbf{r}_2,\mathbf{r}_3</math> is unknown, they can be approximated using [[linear interpolation]].  Barycentric coordinates provide a convenient way to compute this interpolation.  If <math>\mathbf{r}</math> is a point inside the triangle with barycentric coordinates <math>\lambda_1</math>, <math>\lambda_2</math>, <math>\lambda_3</math>, then

<math display=block>f(\mathbf{r}) \approx \lambda_1 f(\mathbf{r}_1) + \lambda_2 f(\mathbf{r}_2) + \lambda_3 f(\mathbf{r}_3)</math>

In general, given any [[unstructured grid]] or [[polygon mesh]], this kind of technique can be used to approximate the value of {{mvar|f}} at all points, as long as the function's value is known at all vertices of the mesh.  In this case, we have many triangles, each corresponding to a different part of the space.  To interpolate a function {{mvar|f}} at a point <math>\mathbf{r}</math>, first a triangle must be found that contains <math>\mathbf{r}</math>.  To do so, <math>\mathbf{r}</math> is transformed into the barycentric coordinates of each triangle.  If some triangle is found such that the coordinates satisfy <math>0 \leq \lambda_i \leq 1 \;\forall\; i \text{ in } 1,2,3</math>, then the point lies in that triangle or on its edge (explained in the previous section). Then the value of <math>f(\mathbf{r})</math> can be interpolated as described above.

These methods have many applications, such as the  [[finite element method]] (FEM).