Grassmann's laws describe empirical results about how the perception of mixtures of colored lights (i.e., lights that co-stimulate the same area on the retina) composed of different spectral power distributions can be algebraically related to one another in a color matching context. Discovered by Hermann Grassmann^[1] these "laws" are actually principles used to predict color match responses to a good approximation under photopic and mesopic vision. A number of studies have examined how and why they provide poor predictions under specific conditions.^[2][3]

Modern interpretation

Grassmann expressed his first law with respect to a circular arrangement of spectral colors in this 1853 illustration.^[4]

The four laws are described in modern texts^[5] with varying degrees of algebraic notation and are summarized as follows (the precise numbering and corollary definitions can vary across sources^[6]):

These laws entail an algebraic representation of colored light.^[7] Assuming beam 1 and 2 each have a color, and the observer chooses ${\displaystyle (R_{1},G_{1},B_{1})}$ as the strengths of the primaries that match beam 1 and ${\displaystyle (R_{2},G_{2},B_{2})}$ 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 ${\displaystyle (R,G,B)}$, where:

${\displaystyle R=R_{1}+R_{2}\,}$

${\displaystyle G=G_{1}+G_{2}\,}$

${\displaystyle B=B_{1}+B_{2}\,}$

Grassmann's laws can be expressed in general form by stating that for a given color with a spectral power distribution ${\displaystyle I(\lambda )}$ the RGB coordinates are given by:

${\displaystyle R=\int _{0}^{\infty }I(\lambda )\,{\bar {r}}(\lambda )\,d\lambda }$

${\displaystyle G=\int _{0}^{\infty }I(\lambda )\,{\bar {g}}(\lambda )\,d\lambda }$

${\displaystyle B=\int _{0}^{\infty }I(\lambda )\,{\bar {b}}(\lambda )\,d\lambda }$

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

See also

• Color space
• CIE 1931 color space

References