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) * Zn
Following this, a matrix (matrix elements Mij)
[M11 M12 M13] --> gives R value
M = [M21 M22 M23] --> gives G value
[M31 M32 M33] --> gives B value
for 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.
The only thing I had to change was the equations; you apparently placed the parameters a little bit wrong, such that the color space does not show all the colors which it should.
The things I changed are
– Schmantii Oct 10 '19 at 20:07myR(\u,\v,\w),myG(\u,\v,\w),myB(\u,\v,\w)(instead of\u,\u,\v),xr(\u,\v)(instead of\u,\u),zr(\u,\w)(instead of\u,\v) andmyR(z,x,y)(instead ofz,x,z), which then prints the correct mapping (currently using samples between 30 and 40, which does not need so much compiling time).But just because I'm curious: Do you have any explanation or idea why
– Schmantii Oct 10 '19 at 20:10\L(=\u),\a(=\v) and\b(=\w) did not work? Internal processing methods ofpgfplots?\a,\band\Lis the culprit but would assume it is\L. It is generally not recommended to play with one-letter uppercase letters for macros, some of them even get redefined by the engine, e.g.xlatexuses\U. But I really don't know.) I plan to fix the redefinitions when I have more time. – Oct 10 '19 at 20:12Thanks again! And take your time - I will check again after your update, such that other people searching for this (CIELab is a common model used in material sciences, lighting engineering or other fields dealing with coloring / color changes or similar) later will have the correct model and data to work with.
– Schmantii Oct 10 '19 at 20:21The space is intended to (somewhat) look like a sphere, such that I also changed
– Schmantii Oct 10 '19 at 20:29zmaxfrom100to120and defined the x- and y-coordinate to be scaled with50(instead of100). Just for a better (or somewhat more appropriate) appearance, even though (in theory) being not completely correct, since (for example) one (aorb, I forgot) is only fitting to a range of about85, while the other runs up to about115. It's somewhere hidden in the web, but a sphere will at least fit to