Barycentric Coordinates

We have already seen how we can represent any point P in an affine space by writing it in terms of a frame:

Well, observe that we can use this formula to derive a new representation in terms of points. If we let

Then we can express P by

In this case, the ps are called the barycentric coordinates of P relative to the Qs. Furthermore, notice that the Qs form a set of points, none of which is an affine combination of the others. This follows from the fact that they were constructed from a frame. Such a set is called a simplex. In general, an n-simplex is a set of n + 1 points such that none is an affine combination of the others. Here are canonical examples of simplices in the first few dimensions:

The simplices for higher dimensions become somewhat harder to visualize. The 4-simplex is constructed from five tetrahedra, three meeting at an edge.

Note that vectors can also be represented in barycentric coordinates. Once again, if we express a vector u in a frame by

Then we can rewrite this formula as

We now have handy representations for points and vectors in terms of simplices. Note that although these two formulas look very similar, they represent entirely different entities. But it's easy to determine whether a sum represents a point or a vector. If the barycentric coordinates sum to 1, it's a point. If they sum to 0, it's a vector.

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