# Grassmann's law (optics)

For Grassmann's law in linguistics, see Grassmann's law.

In optics, Grassmann's law is an empirical result about human color perception: that chromatic sensation can be described in terms of an effective stimulus consisting of linear combinations of different light colors. It was discovered by Hermann Grassmann.

## Statement

Grassmann expressed his law with respect to a circular arrangement of spectral colors in this 1853 illustration.[1]

An early statement of law, attributed to Grassmann, is:[2]

## Modern interpretation

If a test color is the combination of two other colors, then in a matching experiment based on mixing primary light colors, an observer's matching value of each primary will be the sum of the matching values for each of the other test colors when viewed separately.

In other words, if beam 1 and 2 are the initial colors, 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 $I(\lambda)$ 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.