Grassmann's laws (color science)

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.

• Color space
• CIE 1931 color space