Surface plot with custom color mixing: n-dimensional colormaps

Question

The normal pgfplots-colormap is one-dimensional. It expects a single meta value and infers the corresponding color. However, instead of one meta value I have 3 meta values, each associated with a single value.

For the sake of simplicity let's assume that the three meta values (n1, n2, n3) are always positive and sum up to 1, then the color mixing rule is simple: color = n1 * color1 + n2 * color2 + n3 * color3.

Given this meta data (from a table) I would like to color the mesh with the appropriately mixed color.

Based on this question I have already figured out that the RGB values can be provided explicitly. But I am not able to get the formula right. In particular, I don't know how to access the RBG values of the defined colors.

The following code illustrates the problem:

\documentclass{article}
\usepackage{tikz}
\usepackage{pgfplots}
\pgfplotsset{compat=1.7}

\begin{document}

\definecolor{color1}{red}
\definecolor{color2}{cyan}
\definecolor{color3}{yellow}

\begin{tikzpicture}
  \begin{axis}
  \addplot3[surf, mesh/color input=explicit mathparse,
            point meta/symbolic={\thisrow{n1}, \thisrow{n2}, \thisrow{n3} } % problematic line
            % how do I mix the colors here so that color = n1 * color1 + n2 * color2 + n3 * color3?
           ] table[] {data.csv};
  \end{axis}
\end{tikzpicture}

\end{document}

The data file data.csv contains something like this:

x,y,z,n1,n2,n3
-1.0,-1.0,0,0.243910357114819,0.2666555209251139,0.4894341219600671
-0.5,-1.0,0,0.3013791690628364,0.2152633649632925,0.48335746597387114
0.0,-1.0,0,0.5021192190297844,0.16757485484227622,0.3303059261279394
0.5,-1.0,0,0.5979647407883067,0.14009437103777442,0.2619408881739188

-1.0,-0.5,0,0.164292920913273,0.2655366172800358,0.5701704618066912
-0.5,-0.5,0,0.1355599513531367,0.2040615139888475,0.6603785346580158
0.0,-0.5,0,0.4083574985065873,0.2049743531246625,0.3866681483687502
0.5,-0.5,0,0.4880958726415202,0.2436859445670927,0.2682181827913871

-1.0,0.0,0,0.13994145285914186,0.26041474337409753,0.5996438037667606
-0.5,0.0,0,0.08883801351075463,0.19202791600642005,0.7191340704828253
0.0,0.0,0,0.1290235911491401,0.22620482004695425,0.6447715888039056
0.5,0.0,0,0.1901399752789048,0.49178772678672855,0.31807229793436664

-1.0,0.5,0,0.15879820629696065,0.2619254086260544,0.579276385076985
-0.5,0.5,0,0.11257406503995221,0.18855240164294096,0.6988735333171068
0.0,0.5,0,0.09915126016628552,0.243518553933322,0.6573301859003925
0.5,0.5,0,0.0914288508897056,0.5882997108532986,0.3202714382569958

Answer 1

Welcome to TeX.SE! Here is my solution to this interesting problem:

\begin{filecontents*}{my-data.csv}
x   y   z   n1  n2  n3
0.0 0.0 0.0 0.0 0.0 0.0
1.0 0.0 0.0 1.0 0.0 0.0

0.0 1.0 0.0 0.0 1.0 0.0
1.0 1.0 1.0 0.0 0.0 1.0
\end{filecontents*}

\documentclass[tikz, border=2mm]{standalone}
\usepackage{xparse}
\usepackage{pgfplots}
\usepgfplotslibrary{patchplots}
\pgfplotsset{compat=1.16}

% Each point of the plot will be assigned a color which is a mix of these
% three colors according to the weights given by columns n1, n2 and n3 of the
% data point, respectively.
\definecolor{color1}{rgb}{0.1,1.0,0.5}
\definecolor{color2}{rgb}{0.6,0.2,1.0}
\definecolor{color3}{rgb}{1.0,0.6,0.3}

% Extract the color components into 9 macros so that accessing each component
% when drawing the plot is as fast as possible. The target control sequences
% are \color<num>r, \color<num>g and \color<num>b where <num> is I, II or III
% (corresponding to the three colors 'color1', 'color2' and 'color3').
\ExplSyntaxOn
\int_new:N \l__minze_color_index_int
\seq_new:N \l__minze_color_components_seq

\clist_map_inline:nn { color1, color2, color3 }
  {
    \int_incr:N \l__minze_color_index_int
    \extractcolorspecs {#1} { \l_tmpa_tl } { \l_tmpb_tl }
    \seq_set_split:NnV \l__minze_color_components_seq { , } \l_tmpb_tl

    \clist_map_inline:nn { r, g, b }
      {
        \seq_pop_left:NN \l__minze_color_components_seq \l_tmpa_tl
        \cs_new_nopar:cpx
          { color \int_to_Roman:n { \l__minze_color_index_int } ##1 }
          { \tl_use:N \l_tmpa_tl }
      }
  }
\ExplSyntaxOff

\begin{document}

\begin{tikzpicture}[font=\scriptsize]
\begin{axis}[xlabel=$x$, ylabel=$y$, zlabel=$z$, z label style={rotate=-90}]
  \addplot3 [
    surf, mesh/color input=explicit mathparse,
    patch type=bilinear, shader=interp,
    point meta/symbolic={
      (\thisrow{n1}*\colorIr + \thisrow{n2}*\colorIIr + \thisrow{n3}*\colorIIIr)
        / (\thisrow{n1} + \thisrow{n2} + \thisrow{n3}),
      (\thisrow{n1}*\colorIg + \thisrow{n2}*\colorIIg + \thisrow{n3}*\colorIIIg)
        / (\thisrow{n1} + \thisrow{n2} + \thisrow{n3}),
      (\thisrow{n1}*\colorIb + \thisrow{n2}*\colorIIb + \thisrow{n3}*\colorIIIb)
        / (\thisrow{n1} + \thisrow{n2} + \thisrow{n3})
    },
    ] table[x=x, y=y, z=z] {my-data.csv};

  % Color legend
  \begin{scope}[nodes={minimum width=0.5cm, minimum height=0.25cm,
                       inner sep=0, draw=black, right, label distance=2mm}]
    \coordinate (p) at (rel axis cs:0.3,0.5,1.1);
    \path (p)
      node[fill=color1, label=right:{color 1}] (node 1) {}
      ++(axis direction cs:0,0,-0.15)
      node[fill=color2, label=right:{color 2}] (node 2) {}
      ++(axis direction cs:0,0,-0.15)
      node[fill=color3, label=right:{color 3}] (node 3) {};
  \end{scope}
\end{axis}
\end{tikzpicture}

\end{document}

This conforms to the input data from file my-data.csv:

• The point at (0,0,0) has (n₁, n₂, n₃) = (0,0,0) → black;

• The point at (1,0,0) has (n₁, n₂, n₃) = (1,0,0) → custom color color1;

• The point at (0,1,0) has (n₁, n₂, n₃) = (0,1,0) → custom color color2;

• The point at (1,1,1) has (n₁, n₂, n₃) = (0,0,1) → custom color color3.

Every point of the mesh is assigned a color which is a mix of colors color1, color2 and color3 according to weights respectively given by the n1, n2 and n3 columns for the data point.

It is not necessary to ensure that n1 + n2 + n3 = 1, because my code divides by n1 + n2 + n3 in the appropriate place.

If we take your data file with modified z coordinates (yours are all equal), use col sep=comma in the table options from the \addplot3 [...] table[...] call, and clip=false in the axis options (because this time, I draw the legend outside the axis and I don't want it to be clipped away), we obtain:

\begin{filecontents*}{data.csv}
x,y,z,n1,n2,n3
-1.0,-1.0,0,0.243910357114819,0.2666555209251139,0.4894341219600671
-0.5,-1.0,0,0.3013791690628364,0.2152633649632925,0.48335746597387114
0.0,-1.0,0,0.5021192190297844,0.16757485484227622,0.3303059261279394
0.5,-1.0,0,0.5979647407883067,0.14009437103777442,0.2619408881739188

-1.0,-0.5,0.2,0.164292920913273,0.2655366172800358,0.5701704618066912
-0.5,-0.5,0.2,0.1355599513531367,0.2040615139888475,0.6603785346580158
0.0,-0.5,0.2,0.4083574985065873,0.2049743531246625,0.3866681483687502
0.5,-0.5,0.2,0.4880958726415202,0.2436859445670927,0.2682181827913871

-1.0,0.0,0.9,0.13994145285914186,0.26041474337409753,0.5996438037667606
-0.5,0.0,0.9,0.08883801351075463,0.19202791600642005,0.7191340704828253
0.0,0.0,0.9,0.1290235911491401,0.22620482004695425,0.6447715888039056
0.5,0.0,0.9,0.1901399752789048,0.49178772678672855,0.31807229793436664

-1.0,0.5,1.2,0.15879820629696065,0.2619254086260544,0.579276385076985
-0.5,0.5,1.6,0.11257406503995221,0.18855240164294096,0.6988735333171068
0.0,0.5,1.6,0.09915126016628552,0.243518553933322,0.6573301859003925
0.5,0.5,1.4,0.0914288508897056,0.5882997108532986,0.3202714382569958
\end{filecontents*}

\documentclass[tikz, border=2mm]{standalone}
\usepackage{xparse}
\usepackage{pgfplots}
\usepgfplotslibrary{patchplots}
\pgfplotsset{compat=1.16}

% Each point of the plot will be assigned a color which is a mix of these
% three colors according to the weights given by columns n1, n2 and n3 of the
% data point, respectively.
\definecolor{color1}{rgb}{0.1,1.0,0.5}
\definecolor{color2}{rgb}{0.6,0.2,1.0}
\definecolor{color3}{rgb}{1.0,0.6,0.3}

% Extract the color components into 9 macros so that accessing each component
% when drawing the plot is as fast as possible. The target control sequences
% are \color<num>r, \color<num>g and \color<num>b where <num> is I, II or III
% (corresponding to the three colors 'color1', 'color2' and 'color3').
\ExplSyntaxOn
\int_new:N \l__minze_color_index_int
\seq_new:N \l__minze_color_components_seq

\clist_map_inline:nn { color1, color2, color3 }
  {
    \int_incr:N \l__minze_color_index_int
    \extractcolorspecs {#1} { \l_tmpa_tl } { \l_tmpb_tl }
    \seq_set_split:NnV \l__minze_color_components_seq { , } \l_tmpb_tl

    \clist_map_inline:nn { r, g, b }
      {
        \seq_pop_left:NN \l__minze_color_components_seq \l_tmpa_tl
        \cs_new_nopar:cpx
          { color \int_to_Roman:n { \l__minze_color_index_int } ##1 }
          { \tl_use:N \l_tmpa_tl }
      }
  }
\ExplSyntaxOff

\begin{document}

\begin{tikzpicture}[font=\scriptsize]
\begin{axis}[xlabel=$x$, ylabel=$y$, zlabel=$z$, z label style={rotate=-90},
             clip=false]
  \addplot3 [
    surf, mesh/color input=explicit mathparse,
    patch type=bilinear, shader=interp,
    point meta/symbolic={
      (\thisrow{n1}*\colorIr + \thisrow{n2}*\colorIIr + \thisrow{n3}*\colorIIIr)
        / (\thisrow{n1} + \thisrow{n2} + \thisrow{n3}),
      (\thisrow{n1}*\colorIg + \thisrow{n2}*\colorIIg + \thisrow{n3}*\colorIIIg)
        / (\thisrow{n1} + \thisrow{n2} + \thisrow{n3}),
      (\thisrow{n1}*\colorIb + \thisrow{n2}*\colorIIb + \thisrow{n3}*\colorIIIb)
        / (\thisrow{n1} + \thisrow{n2} + \thisrow{n3})
    },
    ] table[x=x, y=y, z=z, col sep=comma] {data.csv};

  % Color legend
  \begin{scope}[nodes={minimum width=0.5cm, minimum height=0.25cm,
                       inner sep=0, draw=black, right, label distance=2mm}]
    \coordinate (p) at (rel axis cs:-0.4,0.5,1.1);
    \path (p)
      node[fill=color1, label=right:{color 1}] (node 1) {}
      ++(axis direction cs:0,0,-0.15)
      node[fill=color2, label=right:{color 2}] (node 2) {}
      ++(axis direction cs:0,0,-0.15)
      node[fill=color3, label=right:{color 3}] (node 3) {};
  \end{scope}
\end{axis}
\end{tikzpicture}

\end{document}