How do I calculate the coordinates for secondary colors on a CIE chromacity diagram?

Question

Imagine a typical device gamut on CIE xy diagram:

x and y coordinates for primary colors (red, green and blue) is defined and known, how to calculate secondary colors (cyan, magenta and yellow)?

Answer 1

Performing the operation directly using the chromaticity coordinates (ie: taking the midpoint between the two chromaticity coordinates) will yield incorrect results as the chromaticity diagram is highly non uniform.

Assuming you are using linear light values you could perform the following chain of computations:

• CIE xy to CIE xyY (Optional) for each pair of chromaticity coordinates where the Y Luminance value is retrieved from the sRGB normalised primary matrix. It is important to acknowledge that when you are using chromaticity coordinates alone you don't have any Luminance information which can also produce unexpected results.
• CIE xy to CIE XYZ for each pair of chromaticity coordinates
• CIE XYZ to sRGB for each pair of tristimulus values
• sRGB add of your pair of RGB values
• sRGB to CIE XYZ of you resulting RGB values
• CIE XYZ to CIE xy

Now this is likely more dedicated to SO but you could use colour to perform those computations:

% matplotlib inline

import numpy as np
import pylab

import colour
from colour.plotting import *

# Conversion from RGB to chromaticity coordinates.
# Defining RGB values for reference.
RGB_r = np.array([1, 0, 0])
RGB_g = np.array([0, 1, 0])

# We assume they are encoded in *sRGB* colourspace.
XYZ_r = colour.sRGB_to_XYZ(RGB_r, apply_EOCF=False)
XYZ_g = colour.sRGB_to_XYZ(RGB_g, apply_EOCF=False)

# Conversion to chromaticity coordinates.
xy_r = colour.XYZ_to_xy(XYZ_r)
print(xy_r)
# [ 0.64  0.33]

xy_g = colour.XYZ_to_xy(XYZ_g) 
print(xy_g)
# [ 0.3  0.6]


# Conversion to CIE xyY in order to maintain Luminance ratios.
# Using sRGB Luminance ratios, second row of the NPM.
xyY_r = [0.64, 0.33, colour.sRGB_COLOURSPACE.RGB_to_XYZ_matrix[1, 0]]
xyY_g = [0.3, 0.6, colour.sRGB_COLOURSPACE.RGB_to_XYZ_matrix[1, 1]]

xy_s = colour.XYZ_to_xy(
    colour.sRGB_to_XYZ(
        colour.XYZ_to_sRGB(colour.xyY_to_XYZ(xyY_r), apply_OECF=False) +
        colour.XYZ_to_sRGB(colour.xyY_to_XYZ(xyY_g), apply_OECF=False)))
print(xy_s)
# [ 0.41930366  0.50525886]

# Plotting.
RGB_colourspaces_CIE_1931_chromaticity_diagram_plot(
    ('sRGB', ),
    bounding_box=(-0.1, 0.9, -0.1, 0.9), 
    standalone=False)

pylab.plot(xy_r[0], xy_r[1], 'o', markersize=15, color=RGB_r)
pylab.plot(xy_g[0], xy_g[1], 'o', markersize=15, color=RGB_g)
pylab.plot(xy_s[0], xy_s[1], 'o', markersize=15, color=RGB_s)