## Appendix A: Maxwell's Color Model

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As was mentioned in the main section, Maxwell's color model was described in the paper [James Clerk Maxwell, On the Theory of Compound Colours, and the Relations of the Colours of the Spectrum, Phil. Trans. R. Soc. Lond., 150, pp 57--84, 1860]. Maxwell represented the three primary colors by three points in a plane that are the vertices of a triangle. Let us denote these points by R, G, and B. He selected an origin O that doesn't lie in the plane of the triangle. Arbitrary colors are represented by points in three-space. Each point has an associated vector from the origin to the point. We will use the same symbol to represent a point in three-space and its associated vector. Let C represent some color. The vector associated with C has a length and a direction. The direction is uniquely determined by the point of intersection of the line through O and C with the plane determined by the points R, G, and B. Maxwell called this point in the RGB-plane the quality of the color. The length of the vector relative to the length of the corresponding quality point vector he called the quantity (intensity) of the color. Thus, every point in the RGB-plane has quantity one. The vector associated with any color can be uniquely represented by a linear combination of the linearly independent vectors associated with R, G, and B. Let us now look at how Maxwell represented mixes of colors. In Maxwell's model colors are combined in the same way as force vectors are combined in mechanics. Let P and Q be two points in the plane of R, G, and B (see Figure 14).

Suppose we want to mix $p$ units of P and $q$ units of Q. Let P$^\prime$ and Q$^\prime$ be the points (vectors) given by $$\mathbf{P^\prime}=p\mathbf{P}\qquad\text{and}\qquad \mathbf{Q^\prime}=q\mathbf{Q}. \tag{1}$$ These points will generally not lie in the plane of the triangle. Let T be the vector sum of P$^\prime$ and Q$^\prime$. Define S to be the point of intersection of the line OT and the line PQ, i.e., S is the quality of the vector sum T. Then,

$$\mathbf{T}=\mathbf{P^\prime}+\mathbf{Q^\prime}=p\mathbf{P}+q\mathbf{Q}=\gamma \mathbf{S}\qquad\text{\gamma a scalar}.\tag{2}$$ Thus, $\gamma$ is the quantity of T. Since S lies on the line segment joining P and Q, we have $$\mathbf{S}=\alpha\mathbf{P}+\beta\mathbf{Q}\qquad\text{where \alpha+\beta=1}. \tag{3}$$ Therefore, $$\mathbf{T}=\gamma\mathbf{S}=\gamma\alpha\mathbf{P}+\gamma\beta\mathbf{Q}=p\mathbf{P}+q\mathbf{Q}.\tag{4}$$

Since every point in the plane determined by O, P, and Q has a unique representation as a linear combination of P and Q it follows from equation (4) that

$$\gamma\alpha=p\qquad\text{and}\qquad \gamma\beta=q. \tag{5}$$ Dividing these two relations we obtain $$\frac{\alpha}{\beta}=\frac{p}{q}\qquad\text{or}\qquad \alpha= \frac{p}{q}\,\beta.\tag{6}$$ Combining equation (6) with the relation $\alpha+\beta=1$, we get $$\beta=\frac{q}{p+q}\qquad\text{and}\qquad \alpha=\frac{p}{p+q}.\tag{7}$$ Since $\gamma\alpha=p$, it follows that $\gamma=p+q$. Thus, $$\text{quality of }(p\mathbf{P}+q\mathbf{Q})=\mathbf{S}=\frac{p}{p+q}\,\mathbf{P}+\frac{q}{p+q}\,\mathbf{Q} \tag{8}$$ and $$\text{quantity of }(p\mathbf{P}+q\mathbf{Q})=\gamma=p+q.\tag{9}$$

These are the basic mixing rules for colors.

We have mentioned that any color can be represented by a linear combination of R, G, and B. In particular, any point in the plane of R, G, and B can be represented by a linear combination of the three primaries in which the sum of the multiplier coefficients is one, i.e., any color C in the plane of the triangle has a representation of the form

$$\mathbf{C}=r\mathbf{R}+g\mathbf{G}+b\mathbf{B}\quad\text{with}\quad r+b+g=1.\tag{10}$$

To see this consider a color C that is inside the triangle as shown in Figure 15.

Since C is on the line joining B and D, we have

$$\mathbf{C}=\alpha\mathbf{B}+\beta\mathbf{D}\qquad\text{where \alpha+\beta=1}.\tag{11}$$

Since D lies on the line joining R and G, we have

$$\mathbf{D}=\gamma\mathbf{R}+\delta\mathbf{G}\qquad\text{where \gamma+\delta=1}.\tag{12}$$

combining equations (11) and (12), we get

$$\mathbf{C}=\beta\gamma\mathbf{R}+\beta\delta\mathbf{G}+\alpha\mathbf{B}. \tag{13}$$

Moreover, the sum of the coefficients in equation (13) satisfies

$$\beta\gamma+\beta\delta+\alpha=\beta(\gamma+\delta)+\alpha=\beta+\alpha=1. \tag{14}$$

A similar argument can be made for colors outside the triangle. If C is inside the triangle, the coefficients are all positive. For those colors that lie outside the triangle, at least one of the coefficients is negative.

Maxwell also showed that there was a geometric relation between his method of representing colors and another suggested by Hermann Grassmann. Any point in the Plane determined by R, G, and B can be described in terms of a vector from the point representing white in the triangle to the point in question. The direction of this vector in the plane is related to the hue and the magnitude of this vector is related to the saturation.

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