Photometric Relationships Between Complementary Colors*

Two complementary chromaticities must be additively mixed in the proper brightness ratio in order to produce a neutral mixture. This ratio, L, can be computed from any one of the following expressions. L/y+1/y′=(1+L)/yn,L=(y′/y)·(y−yn)/(yn−y′),L=(y′/y)·(x−xn)/(xn−x′).The symbols, x, y, and x′, y′, are the coordinates of the complementary chromaticities in the I.C.I. chromaticity diagram, in which the point xn, yn represents the chromaticity of the desired “neutral” or gray mixture. The three points (x, y), (xn, yn), and (x′, y′) must be colinear in this order. Neither of the complementaries need be spectrum colors, but the equations may be used to determine the brightness ratios of spectrum complementaries necessary for neutral mixtures. The great brightness ratios resulting from the low luminosity of the blue end of the spectrum are inconvenient for graphical representation, and can be avoided if the energy ratios of complementary spectrum lights necessary for a neutral mixture are computed. Loci can be traced in the I.C.I. chromaticity diagram to give the “moment” of one millilambert of every chromaticity with respect to any assigned neutral stimulus. The brightness ratio of two complementary chromaticities necessary for a neutral mixture is the reciprocal of their moments per millilambert with respect to the neutral. The locus of constant moment (per millilambert), m, is a conic section having the y=0 axis as its directrix, the neutral point as its focus and the moment, m, as its eccentricity. This diagram is of immediate use in the plane vector solution of any problem of additive color mixture. The vector solution of the problem encountered in the projection of additive photographs, the determination of the brightness ratios of the three primaries necessary for a neutral screen, is outlined. The “neutral” chromaticity desired on the screen need not be the neutral with respect to which the moment loci are drawn. The colorimetric purity of any color is the ratio of the moment per millilambert of that color to the moment per millilambert of the spectrum color colinear with the neutral and the sample color. Formulas and tables for the interconversion of colorimetric and excitation purities are given. The dominant wave-lengths and visual efficiencies, in illuminant “C,” of colors having maximum visual efficiencies are plotted as functions of the wave-length limits of the absorption bands, and the complementary relations between such colors are reviewed. Formulas are given for the determination of the chromaticities in illuminant “C” of all colors for which the spectro-phometric curves are linear functions of wave-length. All such colors have the dominant wave-length 580.1 mμ, excitation purity less than 55 percent, or dominant wave-length 480.1 mμ, excitation purity less than 35 percent.