structural:summarization, structural:expansion
Relationship between barycentric and affine coordinates

Barycentric coordinates and affine coordinates are strongly related, and may be considered as equivalent.

In fact, given a barycentric frame

one deduces immediately the affine frame

and, if

are the barycentric coordinates of a point over the barycentric frame, then the affine coordinates of the same point over the affine frame are

Conversely, if

is an affine frame, then

is a barycentric frame. If

are the affine coordinates of a point over the affine frame, then its barycentric coordinates over the barycentric frame are

Therefore, barycentric and affine coordinates are almost equivalent. In most applications, affine coordinates are preferred, as involving less coordinates that are independent. However, in the situations where the important points of the studied problem are affinely independent, barycentric coordinates may lead to simpler computation, as in the following example.