Ternary plot

The advantage of using a ternary plot for depicting chemical compositions is that three variables can be conveniently plotted in a two-dimensional graph. Ternary plots can also be used to create phase diagrams by outlining the composition regions on the plot where different phases exist.

Every point on a ternary plot represents a different composition of the three components.

A parallel to a side of the triangle is the locus of points representing systems with constant chemical composition in the component situated in the vertex opposed to the side.

There are three common methods used to determine the ratios of the three species in the composition.

The first method is an estimation based upon the phase diagram grid. The concentration of each species is 100% (pure phase) in each corner of the triangle and 0% at the line opposite it. The percentage of a specific species decreases linearly with increasing distance from this corner, as seen in figures 3–8. By drawing parallel lines at regular intervals between the zero line and the corner (as seen in the images), fine divisions can be established for easy estimation of the content of a species. For a given point, the fraction of each of the three materials in the composition can be determined by the first.

For phase diagrams that do not possess grid lines, the easiest way to determine the composition is to set the altitude of the triangle to 100% and determine the shortest distances from the point of interest to each of the three sides. By Viviani's theorem, the distances (the ratios of the distances to the total height of 100%) give the content of each of the species, as shown in figure 1.

The third method is based upon a larger number of measurements, but does not require the drawing of perpendicular lines. Straight lines are drawn from each corner, through the point of interest, to the opposite side of the triangle. The lengths of these lines, as well as the lengths of the segments between the point and the corresponding sides, are measured individually. Ratios can then be determined by dividing these segments by the entire corresponding line as shown in the figure 2. (The sum of the ratios should add to 1).

• 
  Figure 1. Altitude method
• 
  Figure 2. Intersection method
• 
  Figure 3. An example ternary diagram, without any points plotted.
• 
  Figure 4. An example ternary diagram, showing increments along the first axis.
• 
  Figure 5. An example ternary diagram, showing increments along the second axis.
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  Figure 6. An example ternary diagram, showing increments along the third axis.
• 
  Figure 7. Empty ternary plot
• 
  Figure 8. Indication of how the three axes work.

Derivation of a ternary plot from Cartesian coordinates

Figure (1) shows an oblique projection of point P(a,b,c) in a 3-dimensional Cartesian space with axes a, b and c, respectively.

If a + b + c = K (a positive constant), P is restricted to a plane containing A(K,0,0), B(0,K,0) and C(0,0,K). If a, b and c each cannot be negative, P is restricted to the triangle bounded by A, B and C, as in (2).

In (3), the axes are rotated to give an isometric view. The triangle, viewed face-on, appears equilateral.

In (4), the distances of P from lines BC, AC and AB are denoted by a′, b′ and c′, respectively.

For any line l = s + t n̂ in vector form (n̂ is a unit vector) and a point p, the perpendicular distance from p to l is

${\displaystyle \left\|(\mathbf {s} -\mathbf {p} )-{\bigl (}(\mathbf {s} -\mathbf {p} )\cdot \mathbf {\hat {n}} {\bigr )}\mathbf {\hat {n}} \right\|\,.}$

In this case, point P is at

${\displaystyle \mathbf {p} ={\begin{pmatrix}a\\b\\c\end{pmatrix}}\,.}$

Line BC has

${\displaystyle \mathbf {s} ={\begin{pmatrix}0\\K\\0\end{pmatrix}}\quad {\text{and}}\quad \mathbf {\hat {n}} ={\frac {{\begin{pmatrix}0\\K\\0\end{pmatrix}}-{\begin{pmatrix}0\\0\\K\end{pmatrix}}}{\left\|{\begin{pmatrix}0\\K\\0\end{pmatrix}}-{\begin{pmatrix}0\\0\\K\end{pmatrix}}\right\|}}={\frac {\begin{pmatrix}0\\K\\-K\end{pmatrix}}{\sqrt {0^{2}+K^{2}+{(-K)}^{2}}}}={\begin{pmatrix}0\\{\frac {1}{\sqrt {2}}}\\-{\frac {1}{\sqrt {2}}}\end{pmatrix}}\,.}$

Using the perpendicular distance formula,

${\displaystyle {\begin{aligned}a'&=\left\|{\begin{pmatrix}-a\\K-b\\-c\end{pmatrix}}-\left({\begin{pmatrix}-a\\K-b\\-c\end{pmatrix}}\cdot {\begin{pmatrix}0\\{\frac {1}{\sqrt {2}}}\\-{\frac {1}{\sqrt {2}}}\end{pmatrix}}\right){\begin{pmatrix}0\\{\frac {1}{\sqrt {2}}}\\-{\frac {1}{\sqrt {2}}}\end{pmatrix}}\right\|\\[10px]&=\left\|{\begin{pmatrix}-a\\K-b\\-c\end{pmatrix}}-\left(0+{\frac {K-b}{\sqrt {2}}}+{\frac {c}{\sqrt {2}}}\right){\begin{pmatrix}0\\{\frac {1}{\sqrt {2}}}\\-{\frac {1}{\sqrt {2}}}\end{pmatrix}}\right\|\\[10px]&=\left\|{\begin{pmatrix}-a\\K-b-{\frac {K-b+c}{2}}\\-c+{\frac {K-b+c}{2}}\end{pmatrix}}\right\|=\left\|{\begin{pmatrix}-a\\{\frac {K-b-c}{2}}\\{\frac {K-b-c}{2}}\end{pmatrix}}\right\|\\[10px]&={\sqrt {{(-a)}^{2}+{\left({\frac {K-b-c}{2}}\right)}^{2}+{\left({\frac {K-b-c}{2}}\right)}^{2}}}={\sqrt {a^{2}+{\frac {{(K-b-c)}^{2}}{2}}}}\,.\end{aligned}}}$

Substituting K = a + b + c,

${\displaystyle a'={\sqrt {a^{2}+{\frac {{(a+b+c-b-c)}^{2}}{2}}}}={\sqrt {a^{2}+{\frac {a^{2}}{2}}}}=a{\sqrt {\frac {3}{2}}}\,.}$

Similar calculation on lines AC and AB gives

${\displaystyle b'=b{\sqrt {\frac {3}{2}}}\quad {\text{and}}\quad c'=c{\sqrt {\frac {3}{2}}}\,.}$

This shows that the distance of the point from the respective lines is linearly proportional to the original values a, b and c.[2]

Cartesian coordinates are useful for plotting points in the triangle. 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 (1/2,√3/2), and the triple (a,b,c) is

${\displaystyle \left({\frac {1}{2}}\cdot {\frac {2b+c}{a+b+c}},{\frac {\sqrt {3}}{2}}\cdot {\frac {c}{a+b+c}}\right)\,.}$

A colorized soil textural triangle from the United States Department of Agriculture

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

Sample	Clay	Silt	Sand	Notes
Sample 1	50%	20%	30%	Because clay and silt together make up 70% of this sample, the proportion of sand must be 30% for the components to sum to 100%.
Sample 2	10%	60%	30%	The proportion of sand is 30% as in Sample 1, but as the proportion of silt rises by 40%, the proportion of clay decreases correspondingly.
Sample 3	10%	30%	60%	This sample has the same proportion of clay as Sample 2, but the proportions of silt and sand are swapped; the plot is reflected about its vertical axis.

• Apparent molar property
• Viviani's theorem
• Barycentric coordinates (mathematics)
• Compositional data
• List of information graphics software
  • Earth sciences graphics software
  • IGOR Pro
  • Origin (data analysis software)
  • Sigmaplot
• Types of ternary plots:
  • Chromaticity diagram
  • de Finetti diagram
  • Flammability diagram
  • Jensen cation plot
  • Piper diagram, used in hydrochemistry
  • QFL diagram
• Project triangle
• Trilemma

• "Excel Template for Ternary Diagrams". serc.carleton.edu. Science Education Resource Center (SERC) Carleton College. Retrieved 14 May 2020.
• "Tri-plot: Ternary diagram plotting software". www.lboro.ac.uk. Loughborough University – Department of Geography / Resources Gateway home > Tri-plot. Retrieved 14 May 2020.
• "Ternary Plot Generator – Quickly create ternary diagrams on line". www.ternaryplot.com. Retrieved 14 May 2020.
• Holland, Steven (2016). "Data Analysis in the Geosciences – Ternary Diagrams developed in the R language". strata.uga.edu. University of Georgia. Retrieved 14 May 2020.

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