Section 35–4 The chromaticity diagram

 (Dotted Triangle / Horseshoe shaped area / Spectral sensitivity curves)

In this section, Feynman discusses the dotted triangle and horseshoe shaped area of a chromaticity diagram as well as spectral sensitivity curves. Alternatively, the section could be titled “The CIE chromaticity diagram (1931 version)” because it was based on CIE 1931 color space in which CIE stands for Commission Internationale de l’Eclairage. The chromaticity diagram is a representation of human color perception developed from the experimental results conducted by John Guild and David Wright. In addition, the CIE defined the standard observer in 1931 using a 2° field of view, which represents an average human’s chromatic response. In 1964, the CIE proposed some improvements, e.g., they defined an additional standard observer using a 10° field of view.

 

1. Dotted Triangle:

“If any one color is represented by Eq. (35.4), we can plot it as a vector in space by plotting along three axes the amounts a, b, and c, and then a certain color is a point. If another color is a′, b′, c′, that color is located somewhere else. The sum of the two, as we know, is the color which comes from adding these as vectors… If we use a blue and a green and a red, as primaries, we see that all the colors that we can make with positive coefficients are inside the dotted triangle, which contains almost all of the colors that we can ever see… (Feynman et al., 1963, p. 35–6).”

 

The dotted triangle (or triangular region) contains the colors that can be produced by mixing three primary colors. Each primary color is typically located at a corner of the triangle. The color equation can be represented by C = aR + bG + cB, in which C is the color to be matched, R, G, B are the chosen primary colors, whereas the chromaticity coordinates a, b, c are the amount of each primary color. One may adjust the brightness of the color C by multiplying the coordinates a, b, c by a constant in which a + b + c = 1. Feynman explains that we can plot any color as a vector in the context of the chromaticity diagram, however, the chromaticity coordinates do not form a vector space in the formal sense. The vector space is a mathematical structure that satisfies certain properties, such as closure under addition and scalar multiplication, among others.

 

Maxwell developed a chart in the form of an equilateral triangle, which represents the relationships between different colors and it approaches pure white at the center. The Maxwell’s color triangle is an idealized model that helps visualize how different colors can be combined in varying proportions to form a wide range of hues (colors dependent on the dominant wavelength). Essentially, any point within the triangle identifies a specific color, and it illustrates the additive color mixing process. The interior of the triangle displays the secondary and tertiary colors that result from combining different proportions of the primary colors, e.g., red and green form yellow, green and blue form cyan, whereas blue and red form magenta (see below). However, this triangle has limitations in representing the full complexity of human color perception, individual variations in color vision, and the intricacies of color mixing in different contexts.

Source: Maxwell's Triangle (appstate.edu)

 

2. Horseshoe shaped area:

“… because all the colors that we can ever see are enclosed in the oddly shaped area bounded by the curve. Where did this area come from? Once somebody made a very careful match of all the colors that we can see against three special ones (Feynman et al., 1963, p. 35–7).”

 

The chromaticity diagram is a plane diagram formed by plotting one of the chromaticity coordinates against another that shows a range of colors. Some colorists prefer the term gamut, which means the range of colors that can be formed by mixing different ratios of primary colors.  The color gamut depicted in a chromaticity diagram, appears in the shape of a horseshoe (instead of saying oddly shaped area). In other words, the chromaticity coordinates of the pure colors in the visible spectral range form a concave curve shaped like the “sole of a shoe.” The horseshoe shaped area is a representation of the limits of human vision and the range of colors that can be perceived by the average human eye. Colors lying outside the horseshoe boundary are imaginary and cannot be produced by any combination of the three primary colors within the visible spectrum.

 

Historically, scientists assumed that three primary colors could be mixed to form all colors, but it was not achievable due to the impurity (or imperfection) of the paints. Furthermore, the original CIE chromaticity diagram is an imperfect system because it is unable to generate the full range of visible (perceptually possible) colors. However, it is possible to “mix” imaginary primary colors and quantify all colors, i.e., these non-real primary colors are defined as lying outside the range of visible colors (see below). Specifically, CIE 1931 XYZ color space (based on imaginary primary colors) was developed from experimental data performed by Wright and Guild in the 1920’s and it serves as a reference for other color spaces. In essence, it is impossible to create all visible colors using real primary colors (e.g., red, green, and blue), however, all visible colors can be mixed using imaginary primary colors that are perceptually impossible.

 

Source: handprint: colormaking attributes

3. Spectral sensitivity curves:

“An example of such experimental results for mixing three lights together is given in Fig. 35–5. This figure shows the amount of each of three different particular primaries, red, green and blue, which is required to make each of the spectral colors. Red is at the left end of the spectrum, yellow is next, and so on, all the way to blue. Notice that at some points minus signs are necessary. It is from such data that it is possible to locate the position of all of the colors on a chart, where the x- and the y-coordinates are related to the amounts of the different primaries that are used (Feynman et al., 1963, p. 35–7).”

 

Fig. 35–5 is described in The Feynman Lectures as “The color coefficients of pure spectral colors in terms of a certain set of standard primary.” Perhaps Feynman could use the term Color Matching Functions (CMFs) or “original Wright and Guild RGB functions,” which are a set of mathematical functions that describe how the human eye perceives different wavelengths of light. In short, the CMFs are the amounts of red, green, and blue light needed to match the standardized intensity of light of a certain wavelength. On the other hand, there are spectral sensitivity curves of three idealized light detectors yielding the CIE tristimulus values X, Y and Z (see below). Feynman explains that at some points minus signs are necessary for RGB CMFs in Fig. 35–5, but this is not necessary for XYZ CMFs that have positive values for all wavelengths, which is an advantage.

Source: CIE 1931 color space - Wikipedia

“That is the way that the curved boundary line has been found. It is the locus of the pure spectral colors. Now any other color can be made by adding spectral lines, of course, and so we find that anything that can be produced by connecting one part of this curve to another is a color that is available in nature (Feynman et al., 1963, p. 35–7).”

 

The curved boundary line in the chromaticity diagram (see below) shows three aspects of color perception: (1) Limits of vision: It delineates the limits of color vision that can be perceived by the human eye under normal lighting conditions. (2) Hue and saturation: It defines the perceivable hues and maximum saturation, while those nearer to the center are less saturated. Saturation refers to “the attribute of color perception that expresses the degree of departure from the gray of the same lightness (Hunter & Harold, 1987, p. 407).” (3) Spectral (monochromatic) locus: It represents the path of pure spectral colors at various wavelengths (as shown below in nanometers) along the visible spectrum. In summary, the Color Matching Functions (CMFs) and curved boundary line are closely linked, as the CMFs guide our understanding of how the three primary colors should be mixed to replicate the perception of colors found along the boundary curve, representing the pure spectral colors visible to human vision.

 

Source: (Rhyne, 2012)

 

Review Questions:

1. Do you agree with Feynman that any color within the dotted triangle can be plotted as a vector in space?

2. How would you explain the oddly shaped area in the chromaticity diagram?

3. How would you explain the curved boundary line in the chromaticity diagram?

 

The moral of the lesson: the horseshoe-shaped area in the chromaticity diagram contains the entire range of colors visible to the human eye, encompassing both pure spectral colors and all the colors that result from combinations of the three primary colors.

 

References:

1. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

2. Hunter, R. S., & Harold, R. W. (1987). The measurement of appearance. New York: John Wiley & Sons.

3. Rhyne, T. M. (2012). Applying color theory to digital media and visualization. Boca Raton: CRC Press.