Spectral matching curves

If the color to be matched is a spectral color of a single wavelength $\lambda'$ with unit energy:

$$L(\lambda)=\delta(\lambda-\lambda')$$

the matching equations become

$$\sum_{j=1}^3 A_j(W) T_j(\lambda') a_{ij} =\int S_i(\lambda)\delta(\lambda-\lambda') d\lambda = S_i(\lambda') \;\;\;(i=1,2,3)$$

Solving these three simultaneous equations we can get the tristimulus values $T_j(\lambda')\;\;\;(i=1,2,3)$ for the given spectral color $\delta(\lambda-\lambda')$. As functions of wavelength $\lambda$, these $T_j(\lambda)$'s are called the spectral matching curves representing amount of each primary needed to match a flat spectral distribution.

The spectral matching curves can also be used to obtain tristimulus values $T_j(L)$ for any given color $L(\lambda)$. To see this, we substitute

$$S_i(\lambda)=\sum_{j=1}^3 A_j(W) T_j(\lambda) a_{ij}$$

into the color matching equations to get

$$\sum_{j=1}^3 a_{ij} A_j(W) T_j(L) =\int S_i(\lambda) L(\lam... ...um_{j=1}^3 A_j(W) a_{ij} \int T_j(\lambda) L(\lambda) d\lambda$$

As this equation holds for all three types of cells (i=1,2,3), we see that

$$T_j(L)=\int L(\lambda)T_j(\lambda) d\lambda\;\;\;\;(j=1,2,3)$$

the tristimulus values for matching any color $L(\lambda)$ can be easily obtained from the spectral matching curves $T_j(\lambda)$.