Revisiting the Geometry of a Ternary Diagram with the Half-Taxi Metric

An alternative definition of distance is presented for observations plotted in a ternary diagram and, more generally, for observations in a compositional data set. This definition, which conforms to the triangular coordinate system of the ternary diagram, is compared to other distance measures, and is shown to be tied to the covariance structure of compositional data. Angular differences are also discussed briefly in an Appendix.