I have a question: what is the relation between the barycentric and the projective coordinates? Are the first one a particular case of the second? Thank!
Edit: the setting is the plane, and in particular I am asking if the barycentric coordinates on a triangle are equivalent or similar to the projective coordinates in the plan where the vertex of the triangle are labeled with $[0:0:1],[0:1:0],[1:0:0]$.