Ternary plot

A ternary plot, ternary graph, triangle plot, simplex plot, or de Finetti diagram is a barycentric plot on three variables which sum to a constant. It graphically depicts the ratios of the three variables as positions in an equilateral triangle. It is used in petrology, mineralogy, metallurgy, and other physical sciences to show the compositions of systems composed of three species. In population genetics, it is often called a de Finetti diagram. In game theory, it is often called a "simplex plot".

In a ternary plot, the proportions of the three variables "a", "b", and "c" must sum to some constant, "K". Usually, this constant is represented as 1.0 or 100%. Because "a" + "b" + "c" = "K" for all substances being graphed, any one variable is not independent of the others, so only two variables must be known to find a sample's point on the graph: for instance, "c" must be equal to "K" − "a" − "b". Because the three proportions cannot vary independently, it is possible to graph the intersection of all three variables in only two dimensions.

A ternary plot is drawn as triangle. Each base, or side, of the triangle represents a proportion of 0%, with the point of the triangle opposite that base representing a proportion of 100%. As a proportions increases in any one sample, the point representing that sample moves from the base to the opposite point of the triangle.

In Cartesian coordinates

In Cartesian coordinates, consider an equilateral ternary plot where $a=100\%$ is placed at $(x,y)=(0,0)$ and $b=100\%$ at $(1,0)$. Then $c=100\%$ is $(frac\{1\}\{2\},frac\{sqrt\{3\{2\})$, and the triple $(a,b,c)$ is $(b+frac\{c\}\{2\},frac\{sqrt\{3\{2\}c)$

Example

This example shows how this works for a hypothetical set of three soil samples:

Plotting the points

External links

* [http://www.uwgb.edu/dutchs/petrolgy/Teutect.htm Diagrams of ternary eutectic systems]