graphics - How to conveniently plot 3-category Dirichlet data in equilateral triangle instead of 2-simplex

I'm wondering if there's something in Mathematica similar to the built-in function in R shown in the figures below, borrowed from this post, possibly with flexible axes "orientation", ticksmarks, tick numbers, and gridlines.

By a 3-category Dirichlet distribution, it means that each data point is in the form of $\{u, v, 1-u-v\}$, where the degree of freedom is two with $0

Currently I've been doing something like the demonstrative code below, transforming the data points myself from $\{u,v\}$ in the usual Cartesian coordinates to the "triangular" coordinates. (here the "vertical" axis is flipped just like those plots from R)

ClearAll[Opt, data, dN]; 
dN = 100; 
data = RandomReal[{0, 1}, {dN, 3}]; 
data = data/(Total /@ data); 
Opt = {PlotStyle -> PointSize -> Medium, AspectRatio -> 1, PlotRange -> {{0, 1}, {0, 1}}, GridLines -> {{1}, {1}} };

GraphicsRow[{ListPlot[data[[;; , 1 ;; 2]], 
    Epilog -> Line@{{1, 0}, {0, 1}}, Evaluate@Opt] , 
ListPlot[ Thread@{1/2 (1 + data[[;; , 1]] - data[[;; , 2]]), Sqrt[3] data[[;; , 3]]/2} , 
    Epilog -> {FaceForm[], EdgeForm@Thickness@.01, Triangle@{{0, 0}, {1, Sqrt[3]}/2, {1, 0}}}, Evaluate@Opt]}, ImageSize -> 500]

Firstly I feel kind of stupid having to do it this way every time. Secondly, it's tedious to add the tickmarks, gridlines, etc.

So, repeating my question statement in the opening line:

Is there actually a similar built-in graphics package in MMA? If not, is there a convenient way to achieve some if not all the features in a "triangular plot" shown in the R plots?

I would imagine that Dirichlet distribution is pretty common and someone have developed something practically useful already.

Pointers to references or any suggestions will be appreciated.

Answer

How is this? It does not support all Graphics options, but that can be customized. As is, it mimics the styling of ListPlot.

ClearAll[BarycentricPlot];
BarycentricPlot[data_?MatrixQ,
  OptionsPattern[{
    "Ticks" -> N@Range[0, 1, 1/10]
    }]] := 

 Module[{λ, pts, plot, h, c, opts, g, s, prolog, gridlinesx, 
   gridlinesy, ticks},
  h = Sin[Pi/3];
  c = {1/2, h/3};
  λ = data/Total[data, {2}];
  plot = ListPlot[λ.Developer`ToPackedArray[
      N[{{0, 0}, {1, 0}, {1/2, h}}]]];
  opts = Options[plot];
  ticks = OptionValue["Ticks"];
  gridlinesy = ticks[[2 ;; -2]] h;

  gridlinesx = gridlinesy/Tan[Pi/3];
  g[label_, θ_, ϕ_] :=
   Graphics[{
     Rotate[
      Text[Style[label, {}], {1/2, h + 0.1}], ϕ, {1/2, 
       h + 0.1}],
     GridLinesStyle /. opts,
     Line@Transpose[{
        Transpose[{gridlinesx , gridlinesy}],
        Transpose[{1 - gridlinesx , gridlinesy}]

        }]
     },
    PlotRangePadding -> 0,
    ImageMargins -> 0.1,
    PlotRange -> {{0, 1}, {0, 2 h}},
    Axes -> {True, False},
    Ticks -> {Table[{x, Rotate[x, 4 Pi/3 + θ]}, {x, ticks}], 
      None},
    AxesStyle -> (AxesStyle /. opts)
    ];

  s = 1.055;
  prolog = Graphics[{
     Inset[g["\!\(\*SubscriptBox[\(μ\), \(3\)]\)", -Pi, 0], c, c, 
      s],
     Rotate[
      Inset[g["\!\(\*SubscriptBox[\(μ\), \(1\)]\)", 0, Pi], c, c, 
       s], 2/3 Pi, c],
     Rotate[
      Inset[g["\!\(\*SubscriptBox[\(μ\), \(2\)]\)", -Pi, -Pi], c, 
       c, s], 4/3 Pi, c]

     },
    PlotRange -> {{0, 1}, {0, h}},
    PlotRangePadding -> Scaled[0.15],
    Frame -> False
    ];
  Show[{prolog, plot}]
  ]

dN = 1000;
data = RandomReal[{0, 1}, {dN, 3}];

BarycentricPlot[data]