Grassmann’s laws in color science

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 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.

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.

Each color impression can be completely described with exactly three basic sizes.

Mathematical notation: $\{C\}\equiv R.\{R\}+G.\{G\}+B.\{B\}$ bzw. $\{C\}\equiv {\begin{pmatrix}R\\G\\B\end{pmatrix}}$ in alternative spelling.
Graßmann himself likes to use the three basic quantities of basic color (spectral color), color intensity and white intensity. Today, this trinity is called the HSV color space and is modeled as a cone in the adjacent picture; the abbreviations stand for Hue (hue), Saturation (saturation) and Value of Lightness (also Brightness or Luminance, German darkness). The law is also applicable to three primary colors (such as the CIE primary valences or RGB) – just three colors, each of which can not be made by a mixture of the other two.

Second law: The appearance of a mixture 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.

Grassman’s second law of additive color mixture.png
If one mixes a color with a changing hue with a color in which the hue always remains the same, then colors with changing hue emerge, as illustrated by the intersections of the color surfaces in the accompanying picture.

Mathematical notation:
Two colors, $\{C_{1}\}\equiv {\begin{pmatrix}R_{1}\\G_{1}\\B_{1}\end{pmatrix}}$ after additive color blending. $\{C\}\equiv \{C_{1}\}+\{C_{2}\}\equiv {\begin{pmatrix}R_{1}+R_{2}\\G_{1}+G_{2}\\B_{1}+B_{2}\end{pmatrix}}$
Hereby, Graßmann basically describes the (mathematical) homogeneity of the color space – no matter which color change on a color, the mixed product follows analogously.

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.

Grassman’s third law of additive color mixture.png
The hue of a color resulting from additive color mixing depends only on the color impression of the starting colors, but not on their physical (spectral) compositions. The picture on the right demonstrates the formation of two mutually more metameric colors (M1 and M2) from different color components (K1¹, K1² and K1³ or K2¹, K2² and K2³).

Mathematical notation: $\{C'\}\equiv k.\{C\}\equiv {\begin{pmatrix}k.R\\k.G\\k.B\end{pmatrix}}$
This law states that the mixing behavior of even the metameric colors – ie those with the same color impression but at the same time different spectral composition – can be described exactly on the basis of their color impression. Conversely, no direct conclusions about the spectral composition of a color can be drawn from the mixing behavior.

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.

Grassman’s fourth law of additive color mixture.png
The intensity (or total intensity) of an additive mixed color (T3) corresponds to the sum of the intensities of the output colors
(in the scheme limited to T1 and T2).

Mathematical notation: $T(A+B)\equiv T(A)+T(B)$ (with T as correspondence the total intensity or luminance of a color impression)
According to David L. MacAdam, this law applies only to the special case of an idealized, one-point reduced source, but not to more expansive color surfaces. Graßmann had only dealt with the special case mentioned above.

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:

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

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

$B=B_{1}+B_{2}\,$
Grassmann’s law can be expressed in general form by stating that for a given color with a spectral power distribution {\displaystyle I(\lambda )} 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.

importance
The postulates do not apply universally to all seeing beings, but especially to the human visual sense. The laws specify the general significance of trichromaticity. They make it possible to make accurate predictions about the expected equality impression of colors and thus form the basis of the colorimetry, with the help of which, for example, the color reproduction in print or the reproduction on monitors is standardized. In general, this teaching of color designations allows a description of color valence by graphic means, as illustrated in the image on the right of a graph of Graßmann’s color mixing calculation using vectors. This type of calculation is also fundamentally based on Graßmann’s work.

First publication
When Hermann Ludwig Ferdinand von Helmholtz developed his three-color theory around 1850 on the basis of an older theory of color perception by Thomas Young, this was noticed by numerous nineteenth-century scientists. Graßmann based his considerations on theories of Sir Isaac Newton, which he had developed in his work “Opticks: or, a treatise on the reflections, refractions, inflexions and colors of light” (London 1704).

In dealing with some erroneous conclusions of Helmholtz (1852), which corrected this after the appearance of Graßmann’s work, Graßmann clarified the color theory of Newton and this precisely refined in terms of a description in a color space. In February 1853 he published an article in “Poggendorff’s Annals of Physics and Chemistry”

The book titled “The Theory of Color Mixing” begins with the following words:

“Mr. Helmholtz shares a series of partly new and ingenious observations, from which he concludes that the theory of color mixing generally accepted since Newton is erroneous in the most essential points, and namely that there are only two prismatic colors, yellow and indigo , which deliver mixed white. Therefore, it would not be superfluous to show how Newton’s theory of color mixing reaches to a certain point, and especially the proposition that every color has its complementary color, which yields white to it mixed with it, from undeniable facts with mathematical evidence so that this sentence must be considered as one of the most well-founded in physics. I will then show how the positive observations made by Helmholtz, instead of testifying against this theory, can serve to confirm it, partly to supplement it. ”
He gives his “laws of color mixing” the following wording:

1. (…) “every color impression [decomposes] (…) into three mathematically determinable moments (…): the color tone, the intensity of the color, and the intensity of the blended white.”
2. (…) “if one of the two lights to be mixed changes one thing continuously (…), the impression of the mixture also changes constantly.”
3. There are “(…) two colors, each of which has a constant hue, constant intensity of color, and constant intensity of the admixed white, as well as a constant mixture of colors (…), no matter from which homogeneous colors those are composed.”
4. (…) “the total light intensity of the mixture the sum (…) of the intensities of the mixed lights.”

Grassman color circle 1853.png
For illustration, he added various graphical representations, as the adjacent figure shows by way of example. Using this geometric representation of the relationships at the color level, he describes a particular mix of proportions of colors A and B using the following definitions and terms:

A and B are homogeneous colors, O is the white point;
D represents the maximum saturation and the color point C corresponds to the hue in its severity.
(a + b) OC stands for the intensity of the color components.
(a + b) CD represents the intensity of the white component.
(a + b) OD (with OD = 1) expresses the total intensity.

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