Aternary plot,ternary graph,triangle plot,simplex plot, orGibbs triangle is abarycentricplot on three variables whichsum to a constant.[1] It graphically depicts the ratios of the three variables as positions in anequilateral triangle. It is used inphysical chemistry,petrology,mineralogy,metallurgy, and other physical sciences to show the compositions of systems composed of three species. Ternary plots are tools for analyzingcompositional data in the three-dimensional case.
In a ternary plot, the values of the three variablesa,b, andc must sum to some constant,K. Usually, this constant is represented as 1.0 or 100%. Becausea +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 toK −a −b. Because the three numerical values cannot vary independently—there are only twodegrees of freedom—it is possible to graph the combinations of all three variables in only two dimensions.
The advantage of using a ternary plot for depictingchemical compositions is that three variables can be conveniently plotted in a two-dimensional graph. Ternary plots can also be used to createphase diagrams by outlining the composition regions on the plot where different phases exist.
There are three equivalent methods that can be used to determine the values of a point on the plot:
Parallel line orgrid method. The first method is to use a diagram grid consisting of lines parallel to the triangle edges. A parallel to a side of the triangle is the locus of points constant in the component situated in the vertex opposed to the side. Each component is 100% in a corner of the triangle and 0% at the edge opposite it, decreasing linearly with increasing distance (perpendicular to the opposite edge) from this corner. By drawing parallel lines at regular intervals between the zero line and the corner, fine divisions can be established for easy estimation.
Perpendicular line oraltitude method. For diagrams that do not possess grid lines, the easiest way to determine the values is to determine the shortest (i.e. perpendicular) distances from the point of interest to each of the three sides. ByViviani's theorem, the distances (or the ratios of the distances to thetriangle height) give the value of each component.
Corner line orintersection method. The third method does not require the drawing of perpendicular or parallel 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. The ratio of the measured lines then gives the component value as a fraction of 100%.
A displacement along a parallel line (grid line) preserves the sum of two values, while motion along a perpendicular line increases (or decreases) the two values an equal amount, each half of the decrease (increase) of the third value. Motion along a line through a corner preserves the ratio of the other two values.
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.
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.
Unlabeled triangle plot with major grid lines
Unlabeled triangle plot with major and minor grid lines
Ifa +b +c =K (a positive constant),P is restricted to a plane containingA(K,0,0),B(0,K,0) andC(0,0,K). Ifa,b andc each cannot be negative,P is restricted to the triangle bounded byA,B andC, as in (2).
In (3), the axes are rotated to give anisometric view. The triangle, viewed face-on, appearsequilateral.
In (4), the distances ofP from linesBC,AC andAB are denoted bya′,b′ andc′, respectively.
For any linel =s +tn̂ in vector form (n̂ is a unit vector) and a pointp, theperpendicular distance fromp tol is
In this case, pointP is at
LineBC has
Using the perpendicular distance formula,
SubstitutingK =a +b +c,
Similar calculation on linesAC andAB gives
This shows that the distance of the point from the respective lines is linearly proportional to the original valuesa,b andc.[4]
Analogue on a Cartesian grid by adding lines of slope −1. The scale of thec axis is that of thea andb axes. The cross denotes the pointa =b =c.
Cartesian coordinates are useful for plotting points in the triangle. Consider an equilateral ternary plot wherea = 100% is placed at(x,y) = (0,0) andb = 100% at(1,0). Thenc = 100% is and the triple(a,b,c) is
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.
