Our expression for the shape functions is, in fact, a transformation from
cartesian coordinates to a natural set of coordinates, often called *
area coordinates* or * barycentric coordinates*. If we look at the
transformation for for a point within the element with coordinates
, we note that,

which is just a ratio of the areas of the triangles and .

Thus we have a transformation from a point inside an element in terms of its barycentric coordinates to the global cartesian system,