# Grassmann's laws (color science)

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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]):

 First law: Two colored lights appear different if they differ in either dominant wavelength, luminance or purity. Corollary: For every colored light there exists a light with a complementary color such that a mixture of both lights either desaturates the more intense component or gives uncolored (grey/white) light. Second law: The appearance of a mixture of light made from two components changes if either component changes. Corollary: A mixture of two colored lights that are non-complementary result in a mixture that varies in hue with relative intensities of each light and in saturation according to the distance between the hues of each light. Third law: There are lights with different spectral power distributions but appear identical. First corollary: such identical appearing lights must have identical effects when added to a mixture of light. Second corollary: such identical appearing lights must have identical effects when subtracted (i.e., filtered) from a mixture of light. Fourth law: The intensity of a mixture of lights is the sum of the intensities of the components. This is also known as Abney's law.

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.