buddydanax.blogg.se - Reading a ternary diagram

Science Education Resource Center (SERC) Carleton College.

"Excel Template for Ternary Diagrams".

^ Karl Tuyls, "An evolutionary game-theoretic analysis of poker strategies", Entertainment Computing January 2009 doi: 10.1016/j.entcom.2009.09.002, p. 9.

"Phylogenetic origin of limes and lemons revealed by cytoplasmic and nuclear markers".

^ Curk, Franck Ollitrault, Frédérique Garcia-Lor, Andres Luro, François Navarro, Luis Ollitrault, Patrick (2016).

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

In (4), the distances of P from lines BC, AC and AB are denoted by a′, b′ and c′, respectively.įor 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 The triangle, viewed face-on, appears equilateral. In (3), the axes are rotated to give an isometric view.

If a, b and c each cannot be negative, P is restricted to the triangle bounded by A, B and C, as in (2). 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). Motion along a line through a corner preserves the ratio of the other two values.ĭerivation of a ternary plot from Cartesian coordinatesįigure (1) shows an oblique projection of point P( a, b, c) in a 3-dimensional Cartesian space with axes a, b and c, respectively. The ratio of the measured lines then gives the component value as a fraction of 100%.Ī 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. The lengths of these lines, as well as the lengths of the segments between the point and the corresponding sides, are measured individually. Straight lines are drawn from each corner, through the point of interest, to the opposite side of the triangle. The third method does not require the drawing of perpendicular or parallel lines. By Viviani's theorem, the distances (or the ratios of the distances to the triangle height) give the value of each component. perpendicular) distances from the point of interest to each of the three sides. For diagrams that do not possess grid lines, the easiest way to determine the values is to determine the shortest (i.e.

Perpendicular line or altitude method.

By drawing parallel lines at regular intervals between the zero line and the corner, fine divisions can be established for easy estimation. 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. 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. The first method is to use a diagram grid consisting of lines parallel to the triangle edges. There are three equivalent methods that can be used to determine the values of a point on the plot:

2 Derivation from Cartesian coordinates.

The values of a point on a ternary plot correspond (up to a constant) to its trilinear coordinates or barycentric coordinates.

Ternary plots can also be used to create phase diagrams by outlining the composition regions on the plot where different phases exist. The advantage of using a ternary plot for depicting chemical compositions is that three variables can be conveniently plotted in a two-dimensional graph. 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. 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. Usually, this constant is represented as 1.0 or 100%. In a ternary plot, the values of the three variables a, b, and c must sum to some constant, K. Ternary plots are tools for analyzing compositional data in the three-dimensional case. In game theory, it is often called a simplex plot. In population genetics, a triangle plot of genotype frequencies is called a de Finetti diagram. It is used in physical chemistry, petrology, mineralogy, metallurgy, and other physical sciences to show the compositions of systems composed of three species. It graphically depicts the ratios of the three variables as positions in an equilateral triangle. Two ternary diagrams with matching axes can be joined at that axis to make a piper diagramĪ ternary plot, ternary graph, triangle plot, simplex plot, Gibbs triangle or de Finetti diagram is a barycentric plot on three variables which sum to a constant.