Grassmann's Law (optics)
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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 distribution In radiometry, photometry (optics), photometry, and color science, a spectral power distribution (SPD) measurement describes the Power (physics), power per unit area per unit wavelength of an illumination (lighting), illumination (radiant exitan ...
s can be algebraically related to one another in a color matching context. Discovered by
Hermann Grassmann Hermann Günther Grassmann (, ; 15 April 1809 – 26 September 1877) was a German polymath known in his day as a linguist and now also as a mathematician. He was also a physicist, general scholar, and publisher. His mathematical work was littl ...
these "laws" are actually principles used to predict color match responses to a good approximation under
photopic Photopic vision is the vision of the eye under well-lit conditions (luminance levels from 10 to 108  cd/m2). In humans and many other animals, photopic vision allows color perception, mediated by cone cells, and a significantly higher vis ...
and mesopic vision. A number of studies have examined how and why they provide poor predictions under specific conditions.


Modern interpretation

The four laws are described in modern texts with varying degrees of algebraic notation and are summarized as follows (the precise numbering and corollary definitions can vary across sources): ;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. Assuming beam 1 and 2 each have a color, 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 \begin R &= R_1 + R_2, \\ G &= G_1 + G_2, \\ B &= B_1 + B_2. \end Grassmann's laws can be expressed in general form by stating that for a given color with a
spectral power distribution In radiometry, photometry (optics), photometry, and color science, a spectral power distribution (SPD) measurement describes the Power (physics), power per unit area per unit wavelength of an illumination (lighting), illumination (radiant exitan ...
I(\lambda) the RGB coordinates are given by \begin 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. \end 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.


See also

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Color space A color space is a specific organization of colors. In combination with color profiling supported by various physical devices, it supports reproducible representations of colorwhether such representation entails an analog or a digital represe ...
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CIE 1931 color space In 1931, the International Commission on Illumination (CIE) published the CIE 1931 color spaces which define the relationship between the visible spectrum and human color vision. The CIE color spaces are mathematical models that comprise a "sta ...


References

{{reflist Color Visual system