Grassmann's law (optics)

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In optics, Grassmann's law is an empirical result about human color perception: chromatic response is (approximately) linear. It was discovered by Hermann Grassmann.

[edit] Statement

If a test color is the combination of two monochromatic colors, then the observer's matching value of each primary will be the sum of the matching values for each of the monochromatic colors when viewed separately.

In other words, if beam 1 and 2 are monochromatic, and the observer chooses $(R 1, G 1, B 1)$ as the strengths of the primaries that match beam 1 and $(R 2, G 2, B 2)$ as the strengths of the primaries that match beam 2, then if the two beams were combined, the matching values will be the sums of the components. Precisely, they will be $(R, G, B)$ , where:

$R= R_1+R_2\,$

$G= G_1+G_2\,$

$B= B_1+B_2\,$

Grassmann's law can be expressed in general form by stating that for a given color with a spectral power distribution $I (λ)$ the RGB coordinates are given by:

$R= \int_0^\infty I(\lambda)\,\bar r(\lambda)\,d\lambda$

$G= \int_0^\infty I(\lambda)\,\bar g(\lambda)\,d\lambda$

$B= \int_0^\infty I(\lambda)\,\bar b(\lambda)\,d\lambda$

Observe that these are linear in I; the functions $\bar r(\lambda), \bar g(\lambda), \bar b(\lambda)$ are the color matching functions with respect to the chosen primaries.