A point in the color space is determined by 3 numbers.
Given 2 colors, i.e. 2 points in the color space, one can define a distance function from them to a number corresponding to the perceived color difference.
One can try various formulas for the color distance and compare them in experiments with humans who must assess which colors are more similar or more different. Such experiments are extremely time consuming and require many test subjects, to average over individual variability.
The problem is that a formula for a very good approximation of the color differences over the entire color space has not been found yet, even if there are many formulas that give acceptable results in certain cases, usually when the color differences are small.
The Euclidean distance does not give a good approximation, so the color space is not Euclidean. The distances corresponding to various Riemannian spaces (i.e. "curved" spaces) give better approximations, but which are still not good enough.
In fact the color distance is not even a distance in the mathematical sense, i.e. it does not satisfy the triangle axiom, so the color space is not a metric space. Perhaps there exists a non-linear transformation from the color space to a metric space.
"Riemannian spaces (i.e. "curved" spaces) give better approximations, but which are still not good enough."
Right, as I said above this work shows we've nonlinear effects at work, whilst this was obvious in the past we should have taken more notice of the fact. Whilst Riemann provides say an ideal mathematical model (the ideal case) as we obserce Nature doesn't always comply, it insists on adding 'distortions' of its own.
For the same reason these 'transfer' curves have nonlinear characteristics it seems other significant interactions are at work, namely cross-coupling between color channels. We see this in color film emulsions where color bleeds from one layer to another and this effect is also nonlinear (if the bleed were linear across the transfer curve then we could simply bias it out but it's not thus in practice it's nigh on impossible to correct).
In effect this cross-coupling is essentially equivalent to cross-modulation distortion in electronic circuits and the mixing can be modeled mathematically in a similar way.
No doubt it remains to be seen whether this new work has any significant bearing on this or not.
Given 2 colors, i.e. 2 points in the color space, one can define a distance function from them to a number corresponding to the perceived color difference.
One can try various formulas for the color distance and compare them in experiments with humans who must assess which colors are more similar or more different. Such experiments are extremely time consuming and require many test subjects, to average over individual variability.
The problem is that a formula for a very good approximation of the color differences over the entire color space has not been found yet, even if there are many formulas that give acceptable results in certain cases, usually when the color differences are small.
The Euclidean distance does not give a good approximation, so the color space is not Euclidean. The distances corresponding to various Riemannian spaces (i.e. "curved" spaces) give better approximations, but which are still not good enough.
In fact the color distance is not even a distance in the mathematical sense, i.e. it does not satisfy the triangle axiom, so the color space is not a metric space. Perhaps there exists a non-linear transformation from the color space to a metric space.