Visual representations of color spaces

A set of three numbers in xyY or XYZ defines a color. 

How does it look? How can it be reproduced?

A good occasion to go back to the roots.

The CIE color system is essentially based on the work of Hermann Graßmann,
one of the greatest scientists in the 19th century [1].
Mostly his four laws are quoted [2], but it boils down to this [3]:
"The whole set of color stimuli constitutes a linear vector space,
named tristimulus space."

Colors behave like vectors in a three-dimensional space. A color is
characterized by three numbers. A fourth number would be redundant
(linearly dependent). A color is not characterized by just one spectrum. For
one color exists an arbitrary number of different spectra, called metamers.
The explanation of Graßmann's laws via spectra [4] is wrong.

Nobody knows what a color 'really' is. Acccording to the philosopher Kant,
we don't know what any object in the space 'really' is. We have just an
impression by our senses, eventually enhanced by instruments.
Colors are described by comparison with reference colors. For length and
weight we need references as well: the meter, the kilogramm.
Reference colors are (for instance) the three CIE primaries, spectral colors
with well defined wavelengths (it's not important, that the first experiments
were executed with a different set). Let's say R,G,B.
According to Graßmann one needs exactly three 'primaries'. This should
have ended a long dispute: three or four?, but it didn't.
Now we can use these spectral colors as base vectors of a coordinate system.
According to Kant, humans have an a-priori idea of space (without any proof),
where we can position objects. By intuition the axes are orthogonal.
Any other color can be described by a linear combination of the three primaries,
according to Graßmann exactly in a vector space. Color matching means:
find the three weights for the primaries to match a given (numerically unknown)
color.
Unfortunately, one needs for the matching of some colors negative weight factors,
which is possible by the concept of vector space, but somewhat disturbing for
real color mixing.
In RGB we can describe a new vector space by base vectors X,Y,Z, which form
a non-orthogonal coordinate system. All colors have non-negative coordinates X,Y,Z
and this construct has the funny feature, that luminance is identical with Y,
whereas the 'imaginary primaries' X and Z don't have luminance. Mathematically
this is not a problem.
The relation between the two spaces is given by a matrix Cxr and its Inverse:
X=(X,Y,Z)'  (column vectors)
R=(R,G,B)'
X=Cxr R
R=Crx X =Cxr^-1 X
Conventionally we draw XYZ as a cartesian coordinate system (with orthogonal axes),
and R,G,B is the set of non-orthogonal base vectors – the arrangement has been swapped.
There is no natural law, which of the coordinate systems has to be shown as a cartesian.

XYZ is universal. Other RGB-systems with new primaries can be added, either as
working spaces like sRGB, aRGB or pRGB (ProPhotoRGB, which uses two non-physical
primaries, which doesn't surprise, because we got used to entirely non-physical primaries
X,Y,Z),  or device RGB systems like monitor spaces. Each RGB system is related to XYZ
using a matrix, thus we can transform as well from one RGB-system to another.

Now it's possible to render a threedimensional visualization of a color space by appropriate
colors, for instance aRGB. But for a monitor these colors would be clipped almost at the

sRGB boundary.
Of course it's not possible to render pRGB correctly, because the blue and green primary
are outside the human gamut, and the red primary would be too dark. Therefore it's wise,
to use pRGB only for regions or real world colors.

There might be still an objection: the reference system contains only spectral colors.
How are the little saturated colors looking? This had been clarified by Graßmann: the vector
addition of any two colors is valid mathematically and by appearance.

These explanations may also help to end the dispute about the 'basic colors', especially
for painting. Many artists have invented their own system. The answer is simple: basic
colors are like primaries. Their location in the XYZ-space defines, which colors can be
created with positive weight factors or mixing ratios. By the way: one shouldn't worry here
about the difference between additive and subtractive color mixing.

One may have serious doubts, whether Graßmann's laws and the whole CIE colorimetry
is really valid, especially considering numerous optical illusions [5] and all these special
effects in color appearance, like Helmholtz-Kohlrausch and other nonlinearities.

It's perhaps surprising, that CIE colorimetry works so very well for image processing and
device calibration for monitors, printers and cameras.

Finally I would like to mention another scientist, which is not as popular as others,
probably because his work is mathematically difficult: Jozef B.Cohen [6].   
His work has been continued by William Thornton, Michael Brill and James Worthey
(for a further search).

Best regards --Gernot Hoffmann

[1] Hermann Günther Graßmann (Grassmann)
http://en.wikipedia.org/wiki/Hermann_Grassmann

[2]
http://de.wikipedia.org/wiki/Gra%C3%9Fmannsche_Gesetze

[3] Claudio Oleari
http://www.slidefinder.net/o/oleari10/32197498/p3
(6th slide).

[4]
http://en.wikipedia.org/wiki/Grassmann%27s_law_%28optics%29
.
[5]
http://www.michaelbach.de/ot/index.html

[6]
Jozef B.Cohen
Visual Color and Color Mixture
http://books.google.de/books?id=W8QeI5di7t4C&pg=PR13&lpg=PR13&dq=cohen+color&source=bl&ots=aYGoI0I99g&sig=IJEgdJjk3yDhi-1NvTD4GdyUIXk&hl=de&sa=X&ei=614hVM_dFsrMyAPbzoDwCA&ved=0CGsQ6AEwCA#v=onepage&q=cohen%20color&f=false