A ternary diagram is a triangular diagram which displays the proportions of three variables
that sum to a constant using
barycentric coordinates in a triangle.
The normalized components 
, 
,
and 
satisfy 
and 
,
and correspond to components 
, 
,
and 
,
respectively. Their values in percent are 
, 
, and 
. The polygon vertices 
, 
, and 
represent 100% of the corresponding component. The first figure
shows grid lines at 
, 
, and 
for 
, 2, ..., 5. In most instances, ternary plots are drawn on
equilateral triangles as in the figure above,
though right triangles are also used (West 2013).
Ternary diagrams are sometimes called ternary plots, triangle plots, ternary graphs, simplex plots, and de Finetti diagrams, though the latter term is usually reserved for a specific family of ternary diagrams commonly studied in population genetics. Such diagrams are encountered often in the study of phase equilibria and appear somewhat often throughout a number of physical sciences.
| point | coordinates  |
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The first figure includes nine boundary points whose barycentric coordinates are listed
in the table above. The triangle centroid 
has barycentric
coordinates 
.
The barycentric coordinates of a point 
in the interior
of the triangle can be obtained graphically as illustrated
above. Let 
,
, and 
be the intersections of the
segments 
, 
, and 
with the rays from 
, 
, and 
through 
, respectively. Then the normalized components are
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
where 
denotes the Euclidean length
of the line segment joining points 
and 
.
A different geometric construction uses segments through 
that are parallel to the sides of the triangle.
Let 
and 
lie on 
,
and 
on 
, and 
and 
on 
, with 
, 
, and 
parallel to 
, 
, and 
, respectively. The normalized components are given by
 |  |  |
(4)
|
 |  |  |
(5)
|
 |  |  |
(6)
|
These ratios apply to any nondegenerate triangle. For an equilateral triangle with unit side length, the denominators are all 1 (West 2013).
Algebraically, the point 
is the affine combination
 |
(7)
|
where 
.
With the Cartesian coordinates 
, 
, and 
used in the figures, this gives
 |
(8)
|
For arbitrary nonnegative component amounts 
, 
, and 
with 
, normalization gives
 |
(9)
|
The normalized triples form the standard 2-simplex in 
, and the map to the plotted triangle
is an affine transformation.
The representation of data as a ternary diagram allows three-component data to be displayed in a two-dimensional plot. In the first figure, grid
lines parallel to 
represent points for which 
is constant.
Similarly, the segments from a polygon
vertex represent data for which the ratio of the other
two components is constant. For example, the ratio 
is fixed along the line
segment 
in the second figure, except at 
, where 
(Cornish).
