The three resulting triangle-count-fractions look like barycentric coordinates in that they sum to 1 and are all between 0 and 1.
3] in the plane, and if we relocate every vertex v from our directed-edge-labeled triangular graph to a new position so that its triple of triangle-count-fractions are the barycentric coordinates with respect to [p.
It turns out that triangle-count-fractions are not the only numerical values that can serve as barycentric coordinates for proper graph drawings.
Every probability measure associated with the sets generated by the 3-partitions provides new generalized barycentric coordinates for the vertices, where each vertex v is associated with the triple of values ([mu]([R.
The keys to the proof that the probability measures provide generalized barycentric coordinates for a proper graph drawing lies in the following two observations:
In other words, if two vertices are adjacent in the triangular graph as they are in Figure 12, then none of the three generalized barycentric coordinates assigned to v can equal the corresponding generalized barycentric coordinate assigned to w.
We now formally define discrete barycentric coordinates (DBCs) for all vertices of a given directed-edge-labeled triangular graph G(V,E) in terms of some given probability measure [mu] applied to the three regions [R.
We will also show in a later section that merely by repositioning vertices according to discrete barycentric coordinates, we have implicitly encoded all of the edge information of our triangular graph.
Before we proceed further with our exposition on generating new kinds of barycentric coordinates, let us review briefly how barycentric coordinates come into play in more traditional settings.
Relation of Cartesian Coordinates to Euclidean Barycentric Coordinates
Cartesian coordinates are used more than barycentric coordinates, but the two have some similarities that are useful to highlight here.
Barycentric coordinates provide a unique location to points within a triangle, and, by natural extension, to every point in the plane.