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A ternary plot, ternary graph, triangle plot, simplex plot, Gibbs triangle or de Finetti diagram is a barycentric
plot Plot or Plotting may refer to: Art, media and entertainment * Plot (narrative), the story of a piece of fiction Music * ''The Plot'' (album), a 1976 album by jazz trumpeter Enrico Rava * The Plot (band), a band formed in 2003 Other * ''Plot' ...
on three variables which sum to a constant. It graphically depicts the ratios of the three variables as positions in an
equilateral triangle In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each othe ...
. It is used in
physical chemistry Physical chemistry is the study of macroscopic and microscopic phenomena in chemical systems in terms of the principles, practices, and concepts of physics such as motion, energy, force, time, thermodynamics, quantum chemistry, statistical mecha ...
,
petrology Petrology () is the branch of geology that studies rocks and the conditions under which they form. Petrology has three subdivisions: igneous, metamorphic, and sedimentary petrology. Igneous and metamorphic petrology are commonly taught together ...
,
mineralogy Mineralogy is a subject of geology specializing in the scientific study of the chemistry, crystal structure, and physical (including optical) properties of minerals and mineralized artifacts. Specific studies within mineralogy include the proces ...
,
metallurgy Metallurgy is a domain of materials science and engineering that studies the physical and chemical behavior of metallic elements, their inter-metallic compounds, and their mixtures, which are known as alloys. Metallurgy encompasses both the sc ...
, and other physical sciences to show the compositions of systems composed of three species. In
population genetics Population genetics is a subfield of genetics that deals with genetic differences within and between populations, and is a part of evolutionary biology. Studies in this branch of biology examine such phenomena as adaptation, speciation, and pop ...
, a triangle plot of genotype frequencies is called a
de Finetti diagram A de Finetti diagram is a ternary plot used in population genetics. It is named after the Italian statistician Bruno de Finetti (1906–1985) and is used to graph the genotype frequencies of populations, where there are two alleles and the popul ...
. In
game theory Game theory is the study of mathematical models of strategic interactions among rational agents. Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs1 Chapter-preview links, ppvii–xi It has appli ...
, it is often called a
simplex In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...
plot.Karl Tuyls, "An evolutionary game-theoretic analysis of poker strategies", ''Entertainment Computing'' January 2009 , p. 9 Ternary plots are tools for analyzing
compositional data In statistics, compositional data are quantitative descriptions of the parts of some whole, conveying relative information. Mathematically, compositional data is represented by points on a simplex. Measurements involving probabilities, proportions, ...
in the three-dimensional case. In a ternary plot, the values of the three variables , , and must sum to some constant, . Usually, this constant is represented as 1.0 or 100%. Because 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, must be equal to . Because the three numerical values cannot vary independently—there are only two degrees 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 depicting
chemical composition A chemical composition specifies the identity, arrangement, and ratio of the elements making up a compound. Chemical formulas can be used to describe the relative amounts of elements present in a compound. For example, the chemical formula for ...
s is that three variables can be conveniently plotted in a two-dimensional graph. Ternary plots can also be used to create
phase diagram A phase diagram in physical chemistry, engineering, mineralogy, and materials science is a type of chart used to show conditions (pressure, temperature, volume, etc.) at which thermodynamically distinct phases (such as solid, liquid or gaseous ...
s by outlining the composition regions on the plot where different phases exist. The values of a point on a ternary plot correspond (up to a constant) to its
trilinear coordinates In geometry, the trilinear coordinates of a point relative to a given triangle describe the relative directed distances from the three sidelines of the triangle. Trilinear coordinates are an example of homogeneous coordinates. The ratio is t ...
or
barycentric coordinates In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related ...
.


Reading values on a ternary plot

There are three equivalent methods that can be used to determine the values of a point on the plot: # Parallel line or grid 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 or altitude 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. By
Viviani's theorem Viviani's theorem, named after Vincenzo Viviani, states that the sum of the distances from ''any'' interior point to the sides of an equilateral triangle equals the length of the triangle's altitude. It is a theorem commonly employed in various ma ...
, the distances (or the ratios of the distances to the triangle height) give the value of each component. # Corner line or intersection 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. File:HowToCalculatePercentCompositions Altitude Method.svg, Figure 1. Altitude method File:HowToCalculate%Compositions Intersection Method.gif, Figure 2. Intersection method File:Ternary.example.1.svg, Figure 3. An example ternary diagram, without any points plotted. File:Ternary plot example, constant lines horizontal.svg, Figure 4. An example ternary diagram, showing increments along the first axis. File:Ternary plot example, constant lines parallel to the right.svg, Figure 5. An example ternary diagram, showing increments along the second axis. File:Ternary plot example, constant lines parallel to the left.svg, Figure 6. An example ternary diagram, showing increments along the third axis. File:Blank ternary plot.svg, Figure 7. Empty ternary plot File:Ternary plot.svg, Figure 8. Indication of how the three axes work. File:Triangle Plot - Major grid lines.svg, Unlabeled triangle plot with major grid lines File:Triangle Plot - Major and minor grid lines.svg, Unlabeled triangle plot with major and minor grid lines


Derivation from Cartesian coordinates

: Figure (1) shows an
oblique projection Oblique projection is a simple type of technical drawing of graphical projection used for producing two-dimensional (2D) images of three-dimensional (3D) objects. The objects are not in perspective and so do not correspond to any view of an ...
of point in a 3-dimensional
Cartesian space A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
with axes , and , respectively. If (a positive constant), is restricted to a plane containing , and . If , and each cannot be negative, is restricted to the triangle bounded by , and , as in (2). In (3), the axes are rotated to give an isometric view. The triangle, viewed face-on, appears
equilateral In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each othe ...
. In (4), the distances of from lines , and are denoted by , and , respectively. For any line in vector form ( is a unit vector) and a point , the
perpendicular distance In geometry, the perpendicular distance between two objects is the distance from one to the other, measured along a line that is perpendicular to one or both. The distance from a point to a line is the distance to the nearest point on that line. Th ...
from to is :\left\, (\mathbf-\mathbf) - \bigl((\mathbf-\mathbf) \cdot \mathbf\bigr)\mathbf \right\, \,. In this case, point is at :\mathbf = \begina\\b\\c\end \,. Line has :\mathbf = \begin0\\K\\0\end \quad\text\quad \mathbf = \frac = \frac = \begin0\\\frac\\-\frac\end \,. Using the perpendicular distance formula, :\begin a' & = \left\, \begin-a\\K-b\\-c\end - \left( \begin-a\\K-b\\-c\end \cdot \begin0\\\frac\\-\frac\end \right) \begin0\\\frac\\-\frac\end \right\, \\
0px PX or px may refer to: In business * PX Index, index of the Prague Stock Exchange * Air Niugini (IATA airline code PX) * Part exchange, a type of contract * Post exchange, a store operated by the Army and Air Force Exchange Service on US Army pos ...
& = \left\, \begin-a\\K-b\\-c\end - \left( 0 + \frac + \frac \right) \begin0\\\frac\\-\frac\end \right\, \\
0px PX or px may refer to: In business * PX Index, index of the Prague Stock Exchange * Air Niugini (IATA airline code PX) * Part exchange, a type of contract * Post exchange, a store operated by the Army and Air Force Exchange Service on US Army pos ...
& = \left\, \begin-a\\K-b-\frac\\-c+\frac\end \right\, = \left\, \begin-a\\\frac\\\frac\end \right\, \\
0px PX or px may refer to: In business * PX Index, index of the Prague Stock Exchange * Air Niugini (IATA airline code PX) * Part exchange, a type of contract * Post exchange, a store operated by the Army and Air Force Exchange Service on US Army pos ...
& = \sqrt = \sqrt \,. \end Substituting , :a' = \sqrt = \sqrt = a\sqrt \,. Similar calculation on lines and gives :b' = b\sqrt \quad\text\quad c' = c\sqrt \,. This shows that the distance of the point from the respective lines is linearly proportional to the original values , and .


Plotting a ternary plot

Cartesian coordinates A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
are useful for plotting points in the triangle. Consider an equilateral ternary plot where is placed at and at . Then is (\frac, \frac), and the triple is :\left(\frac\cdot\frac,\frac\cdot\frac\right) \,.


Example

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


Plotting the points

File:SoilTexture USDA.svg, lang=aa, Plotting Sample 1 (step 1):
Find the 50% clay line File:SoilTexture USDA.svg, lang=ba, Plotting Sample 1 (step 2):
Find the 20% silt line File:SoilTexture USDA.svg, lang=ca, Plotting Sample 1 (step 3):
Being dependent on the first two, the intersect is on the 30% sand line File:SoilTexture USDA.svg, lang=da, Plotting all the samples File:Ternary triangle plot of soil types sand clay and silt.svg, Ternary triangle plot of soil types sand clay and silt programmed with Mathematica


List of notable ternary diagrams

*
Chromaticity diagram Chromaticity is an objective specification of the quality of a color regardless of its luminance. Chromaticity consists of two independent parameters, often specified as hue (h) and colorfulness (s), where the latter is alternatively called ...
*
de Finetti diagram A de Finetti diagram is a ternary plot used in population genetics. It is named after the Italian statistician Bruno de Finetti (1906–1985) and is used to graph the genotype frequencies of populations, where there are two alleles and the popul ...
* Dalitz plot *
Flammability diagram Flammability diagrams show the control of flammability in mixtures of fuel, oxygen and an inert gas, typically nitrogen. Mixtures of the three gasses are usually depicted in a triangular diagram, known as a ternary plot. Such diagrams are availab ...
* Jensen cation plot * Piper diagram, used in hydrochemistry *
QFL diagram A QFL diagram or QFL triangle is a type of ternary diagram that shows compositional data from sandstones and modern sands, point counted using the Gazzi-Dickinson method. The abbreviations used are as follows: * Q - quartz * F - feldspar * L - ...
*
QAPF diagram A QAPF diagram is a double ternary diagram which is used to classify igneous rocks based on mineralogic composition. The acronym QAPF stands for "Quartz, Alkali feldspar, Plagioclase, Feldspathoid (Foid)". These are the mineral groups used for ...
* Classification diagram for
Ultramafic rock Ultramafic rocks (also referred to as ultrabasic rocks, although the terms are not wholly equivalent) are igneous and meta-igneous rocks with a very low silica content (less than 45%), generally >18% MgO, high FeO, low potassium, and are composed ...


See also

*
Apparent molar property In thermodynamics, an apparent molar property of a solution component in a mixture or solution is a quantity defined with the purpose of isolating the contribution of each component to the non-ideality of the mixture. It shows the change in the c ...
*
Viviani's theorem Viviani's theorem, named after Vincenzo Viviani, states that the sum of the distances from ''any'' interior point to the sides of an equilateral triangle equals the length of the triangle's altitude. It is a theorem commonly employed in various ma ...
*
Barycentric coordinates (mathematics) In geometry, a barycentric coordinate system is a coordinate system in which the location of a point is specified by reference to a simplex (a triangle for points in a plane, a tetrahedron for points in three-dimensional space, etc.). The bar ...
*
Compositional data In statistics, compositional data are quantitative descriptions of the parts of some whole, conveying relative information. Mathematically, compositional data is represented by points on a simplex. Measurements involving probabilities, proportions, ...
*
List of information graphics software This is a list of software to create any kind of information graphics: * either includes the ability to create one or more infographics from a provided data set * either it is provided specifically for information visualization Vector graphics V ...
**
Earth sciences graphics software {{More citations needed, date=August 2011 Earth sciences graphics software is a plotting and image processing software used in atmospheric sciences, meteorology, climatology, oceanography and other Earth science disciplines. Earth sciences graphic ...
**
IGOR Pro IGOR Pro is a scientific data analysis software, numerical computing environment and programming language that runs on Windows or Mac operating systems. It is developed by WaveMetrics Inc., and was originally aimed at time series analysis, but ha ...
**
Origin (data analysis software) Origin is a proprietary computer program for interactive scientific graphing and data analysis. It is produced by OriginLab Corporation, and runs on Microsoft Windows. It has inspired several platform-independent open-source clones and alte ...
** R has a dedicated package
ternary
' maintained on the Comprehensive R Archive Network ( CRAN) **
Sigmaplot SigmaPlot is a proprietary software package for scientific graphing and data analysis. It runs on Microsoft Windows. The software can read multiple formats, such as Microsoft Excel spreadsheets, and can also perform mathematical transforms and ...
*
Project triangle The project management triangle (called also the ''triple constraint'', ''iron triangle'' and ''project triangle'') is a model of the constraints of project management. While its origins are unclear, it has been used since at least the 1950s. It ...
*
Trilemma A trilemma is a difficult choice from three options, each of which is (or appears) unacceptable or unfavourable. There are two logically equivalent ways in which to express a trilemma: it can be expressed as a choice among three unfavourable option ...


References


External links

* * * * {{Chemical solutions Diagrams