





3 Main Ideas 
Orthonormal
Color Matching Functions.

Vectorial Sensitivity to narrowband lights. 
Tristimulus vectors,
including amplitude. 
Early
ColorMatching Data of Guild or Wright. 
Thoughts on Matching Data 
The primary
lights are not unique, and the same facts can be
presented in alternate sets of graphs, an awkward
situation. However,

Same Color Matches, Varying Overlap among Functions: 
(x, y) Diagram 
So, we think in terms of the chromaticity diagram...  but
those primary colors are still needed. 
Which Wavelengths Act Most Strongly in Mixtures? 
MacAdam, and later Thornton did
calculations like this. Narrowband lights of constant power are mixed with equal energy light. Some wavelengths perturb the white chromaticity more than others. Thornton coined the term "Prime Colors" for the 3 wavelengths that act most strongly. 
Color Matching
Functions Are in Fact Relatively Stable When the Primary Wavelengths Are Changed. 

Later Thornton found:
Michael
Brill's new theorem:
When
one primary wavelength is changed (say the red
wavelength only) then the associated (red) color
matching function changes only in scale, not in
shape.(For a more leisurely discussion of shifting primary wavelengths, please click here.) 
Prime Colors Must
Relate to the Overlap of Red and Green
Sensitivities... 
Red and green cone functions are
highly overlapping. 
Subtracting green from red gives
peaks to account for Prime Colors, but the scaling
is arbitrary. 
Achromatic sensitivity, ybar,
is a linear combination of red and green. Find a
second combination that is orthogonal to it. 
One Degree of Freedom
Remains, to Remix ω_{1} and ω_{2}, But Keep the Mixtures Orthogonal. 

We start with ω_{1}
and ω_{2}, which are linear combinations of
red and green cone functions, and are orthonormal.
Other orthonormal pairs of functions can be generated
by a rotation matrix: 
Now Make a
Parametric Plot of ω_{2} vs ω_{1}. Bingo, the Shape is Invariant. 
Vectorial
Sensitivity to Wavelength. 
For each λ, the eye's
sensitivity is a vector, (ω_{2}, ω_{1}) . The spectrum locus is the eye's vectorial sensitivity to color. It is not a boundary. The spectrum locus is alternatively the vectorial color of narrowband lights, at unit power. 
Prime colors are the wavelengths where radius is a local maximum! 
Stalking Prime
Colors in the 2 Dimensions of Red and
Green. 
Vectorial
Sensitivity to Wavelength, Now in 3 Dimensions. 
Features of the
Orthonormal System 
Applications 
Color Rendering by
Light Sources 
Oh, Yes,
Calculating Tristimulus Vectors. 
Let L be a light, that is a Spectral
Power Distribution.
The
calculation is essentially the same, but the
benefits of the orthonormal color matching
functions are tremendous!

Issues
Demystified: 
Special Credit 
William A. Thornton Michael H.
Brill (But
MacAdam gets a demerit for disparaging Cohen's
work.) Calculations
were done with Omatrix software. 
William A. Thornton

Jozef
B. Cohen

Background 
General Background,
including Thornton's and Cohen's work is in Render Asking: James A. Worthey, "Color rendering: asking the question," Color Research and Application 28(6):403412, December 2003. Mathematical detail is in Render Calc: James A. Worthey, "Color rendering: a calculation that estimates colorimetric shifts," Color Research and Application 29(1):4356, February 2004. See http://www.jimworthey.com
!

Seldom Asked
Questions (links) 
Stop 
Scroll No Farther 
Material Below Addresses Obscure Questions 
2 colors, C1 and C2
are set to equal power. The mixture is m. 
Cone Sensitivities,
Approximately SmithPokorny Primaries 





















