Graßmann's laws

This article deals with the laws of color theory, for the law of linguistics see Graßmann's Law (Linguistics) .

The grammar school professor , mathematician and Sanskritist Hermann Günther Graßmann (1809–1877) summarized his observations on the description and color mixture of colors in the four Graßmann laws of color theory . The laws established by Graßmann related largely to additive color mixing , but also fundamentally apply to subtractive color mixing .

meaning

The postulates do not apply universally to all seeing living beings, but specifically to the human sense of sight. The laws specify the general meaningfulness of trichromatism . They make it possible to make exact predictions about the expected impression of equality of colors and thus represent the basis of colorimetry , with the help of which, for example, color reproduction in print or display on monitors is standardized. In general, this doctrine of the dimensional designations of the colors allows a description of the color valence with graphic means, as it is shown in the picture on the right on a diagram of the Graßmann color mixture calculation with the help of vectors . This type of calculation also goes back fundamentally to the work of Graßmann.

First publication

When Hermann Ludwig Ferdinand von Helmholtz developed his three-color theory around 1850 on the basis of an older theory by Thomas Young on color perception , it was noticed by numerous scientists of the 19th century. Graßmann based his considerations on theories of Sir Isaac Newton , which the latter 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 he corrected after the publication of Graßmann's work, Graßmann specified Newton's color theory and refined it clearly with regard to a description in a color space. In February 1853 he published an article in " Poggendorffs Annalen der Physik und Chemie ".

The text entitled “On The Theory of Color Mixing” begins with the following words:

"Mr. Helmholtz [shares] a number of partly new and ingenious observations, from which he draws the conclusion that the theory of color mixing generally accepted since Newton is erroneous in the most essential points, and that there are only two prismatic colors, namely yellow and indigo which provide mixed whites. Hence it may not be superfluous to show how Newton's theory of color mixing up to a certain point, and especially the proposition that every color has its complementary color, which when mixed with it produces white, from indisputable facts with mathematical evidence so that this theorem must be regarded 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 rather serve to partly confirm it, partly to supplement it. "

He gives the following wording to his "laws of color mixing" :

1. (...) "each color impression [split] (...) in three mathematically definable moments (...): the hue, the intensity of color and the intensity of the added White."
2. (...) "when speaking of the Both lights to be mixed one constantly changes (...), also the impression of the mixture changes constantly. ”
3. There are“ (...) two colors, each of which has constant color tone, constant color intensity and constant intensity of the added white, also constant Color mixture (...), no matter what homogeneous colors they are composed of. "
4. (...)" The total light intensity of the mixture [is] the sum (...) of the intensities of the mixed lights. "

To illustrate this, he added various graphic representations, as shown in the illustration on the right. With this geometric representation of the relationships on the color plane, he describes a certain mixture 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.

This shows the “purely mathematical” approach on which his thought is based. Graßmann came to the conclusion that with his four laws, all of which were based on experience , he had created a type of representation that could replace Newton's “empirical rule” used up to now.

For many years, the laws established by Graßmann were as good as not recognized in the scientific world - today, however, they form a largely still valid basis for all modern color doctrines and spaces .

First Graßmann law

Every color impression can be completely described with exactly three basic parameters.

Mathematical notation: or in alternative spelling. ${\displaystyle \{C\}\equiv R.\{R\}+G.\{G\}+B.\{B\}}$ ${\displaystyle \{C\}\equiv {\begin{pmatrix}R\\G\\B\end{pmatrix}}}$

Graßmann himself liked to use the three basic quantities basic color ( spectral color ), color intensity and white intensity . Today this trinity is known as the HSV color space and is shown as a cone in the picture opposite; The abbreviations stand for H ue ( hue ), S aturation ( saturation ) and V alue of Lightness (also Brightness or Luminance , German dark level ). The law can also be applied to three primary colors (such as the CIE primary valences or RGB ) - three colors that cannot be produced by mixing the other two.

Second Graßmann law

If you mix a color with a changing hue with a color in which the hue always remains the same, colors with a changing hue are created , as illustrated by the intersections of the colored areas in the accompanying picture.

Mathematical notation:

Two colors, and result in the color after additive color mixing ${\displaystyle \{C_{1}\}\equiv {\begin{pmatrix}R_{1}\\G_{1}\\B_{1}\end{pmatrix}}}$ ${\displaystyle \{C_{2}\}\equiv {\begin{pmatrix}R_{2}\\G_{2}\\B_{2}\end{pmatrix}}}$ ${\displaystyle \{C\}\equiv \{C_{1}\}+\{C_{2}\}\equiv {\begin{pmatrix}R_{1}+R_{2}\\G_{1}+G_{2}\\B_{1}+B_{2}\end{pmatrix}}}$

With this, Graßmann basically describes the (mathematical) homogeneity of the color space - regardless of which hue change you make to a color, the mixed product follows analogously.

Third Graßmann's Law

The hue of a color created by additive color mixing depends only on the color impression of the original colors, but not on their physical (spectral) compositions. The picture on the right demonstrates the development of two metameric colors (M1 and M2) from different color components (K1¹, K1² and K1³ or K2¹, K2² and K2³).

Mathematical notation: ${\displaystyle \{C'\}\equiv k.\{C\}\equiv {\begin{pmatrix}k.R\\k.G\\k.B\end{pmatrix}}}$

This law states that the mixing behavior of even the metameric colors - i.e. those colors with the same color impression but at the same time different spectral composition  - can be precisely described purely 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 Graßmann's law

The intensity (or total intensity) of an additively mixed color (T3) corresponds to the sum of the intensities of the original colors (
limited to T1 and T2 in the scheme).

Mathematical notation: (with T as the equivalent of the total intensity or luminance of a color impression) ${\displaystyle T(A+B)\equiv T(A)+T(B)}$

According to David L. MacAdam , this law only applies to the special case of an idealized source reduced to a point, but not to more extensive areas of color. Graßmann only dealt with the special case mentioned above.

Individual evidence

1. ↑ Günther Wyszecki, WS Stiles, "Color Science: Concepts and Methods, Quantitative Data and Formulas" , ISBN 978-0-471-39918-6
2. ^ H. Grassmann, "On the theory of color mixing," Poggendorffs Annalen 69, 1853, pp. 69–84 . Google Books . Retrieved February 1, 2017.

See also

• Color space
• Opposite color theory
• CIE standard valence system

Web links

• The Grassmann experiment (PDF; 7 MB)
• Seminar paper Daniel Göhring: Hermann Günter Grassmann - Life and Grassmann's Laws (PDF; 645 kB)
• Anna Katharina Hirschmüller, "Investigations into the perception of surface and lighting colors" , (dissertation, PDF; 6 MB)