Analytical Equations to the Chromaticity Cone: Algebraic Methods for Describing Color

TL;DR

This paper develops algebraic equations for the chromaticity cone using affine transformations and homogenization, enabling precise descriptions of color perception and new color order systems.

Contribution

It introduces analytical equations for the chromaticity cone and subsets, facilitating algebraic modeling of color perception and color order systems.

Findings

Derived analytical equations for the chromaticity cone.

Established a method to obtain Macadam ellipses analytically.

Proposed new color order systems based on cone equations.

Abstract

We describe an affine transformation on the (CIE) color matching functions and map the spectral locus as a circle. We then homogenize the right circular cylinder erected by the circle, with respect to a normalizing plane and develop an analytical equation to the chromaticity cone, for the spectral colors. In the interior of the (CIE) chromaticity diagram, by homogenizing elliptic cylinders with respect to the normalizing planes, analytical equations to subsets (also cones) of the chromaticity cone are developed. These equations provide an algebraic method for describing color perception. As an application of the interior chromaticity cones, we demonstrate that by sectioning homogenized cones with planes and projecting, analytical equations to the Macadam ellipses may be derived. Further, the cone equations are used to propose new types of color order systems.