After already posting a question about Drawing a sphere and mapping CIELab color space to it (i.e. generally asking if pgfplots allows an axis-dependent color mapping beforehand), I searched all the calculus and theory stuff about CIELab color space I needed, in order to get the equations I need for a (possible) color mapping.
My idea now is to take the x,y,z values of my sphere (see MWE below), where, for the CIELab Color Space, I consider my sphere coordinates (all calculated in spherical coordinates incl. sin and cos) to be x=a, y=b and z=L.
Starting with CIELab Color Space, the calculation should go as follows: L,a,b --> X,Y,Z --> R,G,B.
Using this, I first defined necessary functions with pgfmanual, p. 1032 / 1033 and
declare function = {<function definitions>}plus ifthenelse (same package)
<statement> ? <yes> : <no> to calculate some coefficients xr,yr,zr which are required to perform L,a,b --> X,Y,Z as follows:
declare function={% Coefficients xr,yr,zr for X,Y,Z calculation xr(\L,\a) = ( (\a/500) + ((\L+16)/116) )^3 > 0.008856 ?% ( ((\a/500) + ((\L + 16) / 116))^3 ) :% ( 116 * ((\a/500) + ((\L + 16) / 116)) - 16 ) / 903.3; yr(\L) = \L > ( 0.008856 * 903.3 ) ?% ((\L + 16)/116)^3 :% \L / 903.3; zr(\L,\b) = ( ((\L + 16) / 116) - (\b/200) )^3 > 0.008856 ?% ((((\L + 16) / 116) - (\b/200))^(3)) :% ( 116 * (((\L + 16) / 116) - (\b/200)) - 16 ) / 903.3; }Doing this, I acquire the X,Y,Z (tristimulus) values by multiplication of those coefficients xr,yr,zr with some illuminant constants Xn,Yn,Zn to achieve
X = xr(\L,\a) * Xn Y = yr(\L) * Yn Z = zr(\L,\b) * ZnFollowing this, a matrix (matrix elements Mij)
[M11 M12 M13] --> gives R value M = [M21 M22 M23] --> gives G value [M31 M32 M33] --> gives B valuefor XYZ --> RGB (both 3x1 vectors) conversion can be used to get the final results (functions R,G,B, depending on \L,\a,\b):
R(\L,\a,\b) = xr(\L,\a) * Xn * M11+ yr(\L) * Yn * M12+ zr(\L,\b) * Zn * M13; G(\L,\a,\b) = xr(\L,\a) * Xn * M21+ yr(\L) * Yn * M22+ zr(\L,\b) * Zn * M23; B(\L,\a,\b) = xr(\L,\a) * Xn * M31+ yr(\L) * Yn * M32+ zr(\L,\b) * Zn * M33;Having this, it should (theoretically / as I would think) be possible to simply plug in the coordinates x=\a [-100;100], y=\b [-100;100] and z=\L [0;100] of my sphere to get the results for each point meta in \addplot3:
\addplot3[point meta = {symbolic = {<R>,<G>,<B>}}] (% Define sphere to be mapped on, incl. limited domains {100*cos(azimuth)*sin(polar)},% x {100*sin(azimuth)*sin(polar)},% y {50*cos(polar)+50}% z );Unfortunately, the CIELab model does not exactly represent a sphere (rather an ellipsoid, since the three values of RGB (color gamut) can not fill the whole color space of XYZ or Lab completely, which is also a pretty nasty companion for every company producing colors in TVs, Smartphones etc.), which is why one also has to place exceptions when mapping RGB to a sphere.
Here, I simply defined the point meta = {symbolic = {<R>,<G>,<B>}} to replace values < 0 --> 0 and > 1 --> 1, in order to prevent the compiling process from crashing (using nested ifthenelse):
point meta={% symbolic={% % R Values in [0;1] ifthenelse( R(z,x,y) < 0 , 0.0 , ifthenelse( R(z,x,y) > 1 , 1.0 , R(z,x,y) ) ),% % G Values [0;1] ifthenelse( G(z,x,y) < 0 , 0.0 , ifthenelse( G(z,x,y) > 1 , 1.0 , G(z,x,y) ) ),% % B Values [0;1] ifthenelse( B(z,x,y) < 0 , 0.0 , ifthenelse( B(z,x,y) > 1 , 1.0 , B(z,x,y) ) )% }% }And here comes the thing: It's not compiling successfully, popping errors like Sorry, an internal routine of the floating point got an ill-formatted floating point number and even (for some reason) Unknown function 'Y' in 'xr(1,Y)' (might be the origin?) which leaves me helpless, since I am lacking the experience to fix it and to exactly know how pgfplots processes the data in mathparse.
Any ideas how to fix this / make it running successfully?
Here's my Code* so far (incl. concrete values plugged in and the commented mesh/colorspace explicit color input=rgb255-option, since I, though having the literature, am not sure (yet) whether R(\L,\a,\b), G(\L,\a,\b) and B(\L,\a,\b) will pop values in [0;1] or [0;255]; I would expect [0;1], though):
\documentclass[tikz,border=3mm]{standalone} \usepackage{pgfplots} \pgfplotsset{compat=1.16} \usepgfplotslibrary{patchplots} \begin{document} \begin{tikzpicture}[% declare function={% % Calculation scheme: Lab --> XYZ --> RGB [0;1] % Math, formulas and values based on % - https://en.wikipedia.org/wiki/Illuminant_D65 % - https://en.wikipedia.org/wiki/CIELAB_color_space % - https://en.wikipedia.org/wiki/Adobe_RGB_color_space % - http://brucelindbloom.com/index.html?Eqn_RGB_XYZ_Matrix.html % - http://www.brucelindbloom.com/index.html?Eqn_Lab_to_XYZ.html % - http://brucelindbloom.com/index.html?Eqn_RGB_XYZ_Matrix.html (with XYZ -> sRGB) % % Declaring Functions to calculate CIE XYZ values --> RGB from the following variables: % \L = L-value of CIE Lab space % \a = a-value of CIE Lab space % \b = b-value of CIE Lab space % xr, yr, zr = Coefficients required to get XYZ (from L,a,b) % % D65 illuminant tristimulus values (T = 6504 K;) % Xn = 0.968774; % Yn = 1.0; % Zn = 1.121774; % % CIE constants (E = epsilon, K = kappa) % E = 0.008856; % K = 903.3; % % XYZ -> sRGB matrix: % [3.2404542 -1.5371385 -0.4985314] --> R Value % M = [-0.9692660 1.8760108 0.0415560] --> G Value % [0.0556434 -0.2040259 1.0572252] --> B Value % % Coefficients xr,yr,zr for X,Y,Z calculation xr(\L,\a) = ( (\a/500) + ((\L+16)/116) )^3 > 0.008856 ?% > E? ( ( (\a/500) + ((\L+16) / 116) )^3 ) :% ( 116 * ((\a/500) + ((\L+16)/116)) - 16 ) / 903.3; % yr(\L) = \L > ( 0.008856 * 903.3 ) ?% < (E * K)? ( (\L+16)/116 )^3 :% \L / 903.3; % zr(\L,\b) = ( ((\L+16)/116) - (\b/200) )^3 > 0.008856 ?% > E? ( ( ((\L+16)/116) - (\b/200) )^3 ) :% ( 116 * (((\L+16)/116) - (\b/200)) - 16 ) / 903.3;% % Calculation of R,G,B via illuminant properties and matrix values (XYZ -> sRGB) R(\L,\a,\b) = xr(\L,\a) * 0.968774 * 3.2404542 + yr(\L) * 1.0 * (-1.5371385) + zr(\L,\b) * 1.121774 * (-0.4985314); % Including Xn, Yn, Zn, first matrix row G(\L,\a,\b) = xr(\L,\a) * 0.968774 * (-0.9692660) + yr(\L) * 1.0 * 1.8760108 + zr(\L,\b) * 1.121774 * 0.0415560; % Including Xn, Yn, Zn, second matrix row B(\L,\a,\b) = xr(\L,\a) * 0.968774 * 0.0556434 + yr(\L) * 1.0 * (-0.2040259) + zr(\L,\b) * 1.121774 * 1.0572252; % Including Xn, Yn, Zn, third matrix row } ] \begin{axis}[axis equal, width = 10cm, height = 10cm, axis lines = center, xmin = -120, xmax = 120, ymin = -120, ymax = 120, zmin = 0, zmax = 100, ticks = none, enlargelimits = 0.3, z buffer = sort, view/h = 45, scale uniformly strategy = units only] \addplot3 [% patch, patch type=bilinear, variable = \azimuth, variable y = \polar, domain = 0:360, y domain = 0:180, fill opacity = 0.5, draw opacity = 1, line width = 0.001 pt, samples = 10, % only for faster compilation mesh/color input=explicit mathparse, % mesh/colorspace explicit color input=rgb255, % if RGB values are calculated in [0 ; 255] point meta={% symbolic={% % R Values [0;1] ifthenelse( R(z,x,y) < 0 , 0.0 , ifthenelse( R(z,x,y) > 1 , 1.0 , R(z,x,y) ) ),% check r < 0 and r > 1 % G Values [0;1] ifthenelse( G(z,x,y) < 0 , 0.0 , ifthenelse( G(z,x,y) > 1 , 1.0 , G(z,x,y) ) ),% check g < 0 and g > 1 % B Values [0;1] ifthenelse( B(z,x,y) < 0 , 0.0 , ifthenelse( B(z,x,y) > 1 , 1.0 , B(z,x,y) ) )% check b < 0 and b > 1 }% }, ] (% Define sphere to be mapped on {100*cos(azimuth)*sin(polar)},% x {100*sin(azimuth)*sin(polar)},% y {50*cos(polar)+50}% z ); \end{axis} \end{tikzpicture} \end{document}Thanks a lot for your ideas! :)- Marius.
*adapted from Schrödinger's cat's proposal in the previous question.