Projective Transformations of Color-Mixture Diagrams*

Projective transformations of a standard coordinate system can be written in the form: u=c3+(e1x+e2y)/(c7x+c8y+1),υ=c3+(e4x+e5y)/(c7x+c8y+1).The shapes and relative sizes of all geometrical figures subjected to such transformations will be invariant if the values of c7, and c8 are invariant, and if the following quantities are also invariant: (e1e5−e2e4)/(e1e2+e4e5),(e22+e52)/(e12+e42).The sizes of such figures are invariant when the numerators and denominators of these expressions are themselves invariant.Colors having equally noticeable differences can be represented by equidistant points in some projective transformation of the standard color-mixture diagram only if in the standard diagram the separations of the points on any line representing a series of equally noticeably different colors are proportional to the square of the distance from some unique point on that line, and if the locus of such unique points for all linear series of colors is a straight line. The locus of colors equally noticeably different from any standard color can be represented adequately in the standard color-mixture diagram by an ellipse. All such ellipses can be transformed by a single projective transformation into equal-sized circles only if the common tangents of every pair of ellipses intersect on a single straight line. This line is the same as the locus of unique points for the series of colors just described. The equation of this straight line is (c7x+c8y+1 = 0), where the constants are those appearing in the denominators of the successful transformation formulas. If the separations of points representing equally noticeably different colors in the standard color-mixture diagram cannot be represented adequately in the manners described, then such colors cannot be represented adequately by equidistant points in any projective transformation of the standard coordinate system.