Applies an affine coordinate trandformation that maps the centroid (1/3,1/3,1/3) to the origin (0,0), and
rescales so that the line segments between (1,0,0), (0,1,0), and (0,0,1) are mapped to segments of length 1.
An input vector v
is first normalized so its component sum to 1 before the map is applied.