Plot a trajectory in a simplex

Question

I would like to plot the make-up of a population over time as it changes its composition. For this, I'd like to make a simplex plot, a classic method in some fields - such as this example from a paper in Biology Letters (Sasaki & Uchida, 2014):

My population consists of three types of entity, so a simplex plot would be a very natural way to visualise it. I have a set of coordinates for the population over time, which it would be easy to plot in an x-by-y graph (if the population consisted of only two types of entity), but I am having some difficulty working out how to plot these on a simplex. I wonder if anyone could advise as to best method? (Perhaps there is even a Mathematica function that I am unaware of.)

Each point in my data consists of three pieces of information: the proportion of each of the three types of entity at that time.

The data looks like this (here's the first few lines):

A         B         C
0.7065    0.2492    0.0443
0.7380    0.2342    0.0278
0.7429    0.2065    0.0506
0.7357    0.1712    0.0931
0.7652    0.1740    0.0608
0.7466    0.1452    0.1082
0.6907    0.1193    0.1900
0.6008    0.0870    0.3122
0.4817    0.0584    0.4599
0.5989    0.0701    0.3310
0.6907    0.0806    0.2287
0.5904    0.0592    0.3504

I would like to plot these points in the simplex and connect them with lines.

Many thanks if you can help.

---

Cited: Sasaki, T. & Uchida, S. 2014. Biol. Lett., 10: 20130903

Answer 1

Here's a start.

Grab some sample data for a trajectory from this excellent python package, and examine it in 3D:

data=Import["https://raw.githubusercontent.com/marcharper/python-ternary/master/sample_data/curve.txt","Table"];
ListPointPlot3D@data

The way the data is constructed, you have a set of triples, where the sum of each triple is 1.0. In a ternary plot, these numbers, which represent fractions of a whole, are represented as the distance from the three corners.

You can define a helper function to go from ternary to cartesian coordinates. From the wiki page,

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 $\left(\tfrac{1}{2},\tfrac{\sqrt{3}}{2}\right)$, and the triple $(a,b,c)$ is $\left(\tfrac{1}{2}\tfrac{2b+c}{a+b+c},\tfrac{\sqrt{3}}{2}\tfrac{c}{a+b+c}\right).$

ternaryToCartesian = {1/2 (2 #2 + #3)/(#1 + #2 + #3), 
    Sqrt[3]/2 #3/(#1 + #2 + #3)} &;

Make a background for the plot,

background = 
  Graphics[{FaceForm[None], EdgeForm[Black], 
    Polygon[{{0, 0}, {1, 0}, {1/2, Sqrt[3]/2}}]}];

and then show the trajectory

Show[
 background,
 Graphics[
  {Blue, Line[ternaryToCartesian @@@ data]}
  ],
 AspectRatio -> 1]

adding in gridlines and tick marks would be an interesting problem - one that has already been solved satisfactorily by the triangleTicks function defined here it turns out:

Show[
 background,
 Graphics@triangleTicks[],
 Graphics[
  {Blue, Line[ternaryToCartesian @@@ data]}
  ],
 AspectRatio -> 1]