If we use triangle-count-fractions as barycentric coordinate locations for the vertices in our directed-edge-labeled triangular graph, we can construct new graph drawings of our abstract triangular graph with an amazing property:
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.
It is fairly easy to imagine how to locate two different points with the same first or second or third barycentric coordinate value, just search along a line parallel to a triangle edge.
Hence, we may treat the first two barycentric coordinates as if they were Cartesian coordinates if all we wish to do is to make some barycentric coordinate graph drawing for some choice of three non-collinear anchor points.
Since the relative measure of the blue region corresponds to the first barycentric coordinate [[alpha].
Finally, if we superimpose our special grid on a map, as shown also in Figure 18, we give ourselves a barycentric coordinate system for the map that is both Euclidean and combinatorial.
The following is the test (based on partial orderings of the barycentric coordinate triple).
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: