Representing Color in a 3D Space

Based on the tristimulus theory, any given color $L(\lambda)$ can be matched by mixing proper amounts of three primaries:

$$L(\lambda) \equiv \sum_{j=1}^3 A_j(L) [P_j(\lambda)]$$

Because the intensities of incoherent lights add linearly, i.e., mixing colors $L_1=\sum_j A_j(L_1) P_j(\lambda)$ and $L_2=\sum_j A_j(L_2) P_j(\lambda)$ results a new color which is the linear combination (weighted sum) of the two colors

$$L_1+L_2=\sum_{j=1}^3 [A_j(L_1)+A_j(L_2)] P_j(\lambda)$$

we can use a three dimensional color space spanned by a set of three primary colors to represent all colors. In this space, each point (or a vector) is a color. And a color $L$ produced by mixing two other colors $L_1$ and $L_2$ is represented by the vector sum of the vectors for $L_1$ and $L_2$.

Any three linearly independent colors can be considered as the primaries $P_j(\lambda),\;\;(j=1,2,3)$ and used as the three bases of the 3D color space. Here linearly independent means none of the three primaries can be written as the linear combination of the other two, i.e.,

$$a_1 P_1(\lambda)+a_2 P_2(\lambda)+a_3 P_3(\lambda) \neq 0$$

unless $a_1=a_2=a_3=0$.

Any given color $L$ in the color space can be specified by its coordinates $\{A_1(L),\;A_2(L),\;A_3(L)\}$, just like a point in a 3D Cartesian coordinate system can be represented by it coordinates $\{ x, y, z\}$ with the following differences

• The color space is a space of functions of $\lambda$, instead of a Cartesian space of geometric points or vectors;
• Orthogonal axes are used in Cartesian coordinate system, while axes used in the color space are usually not orthogonal;
• As a result of the above, different orthogonal axes are related by orthogonal transformations in Cartesian coordinate system while the axes in the color space are related by projective transformation.

Question:

As we know an arbitrary light $L(\lambda)$ is just a function of wavelength $\lambda$. By claiming that all lights can be represented in a 3D color space spanned by a set of three primaries, are we also claiming that all possible functions of $\lambda$ can be represented as a linear combination of three primaries (linearly independent) functions $P_i(\lambda)\;\;(i=1,2,3)$, an obvious wrong mathematical statement?

Fourier expansion...

Answer:

Given the sensitivity functions $S_i(\lambda)$ of the three photoreceptors of a particular visual system, all possible lights as functions of wavelength $L(\lambda)$ are mapped to a 3D color space. Each point in this space represents a set of infinite number of all possible lights, the ``matching colors'', in the sense that they are responded to by the three types of photoreceptors in the visual system in exactly the same way. In other words, each point in the color space is a family of metamers. Now we see one of the reasons why the words light and color should be distinguished.