Color Perception: Opening up the Chromaticity Cone

In the XYZ color space, the subset of the tri-stimuli corresponding to spike-type (monochromatic) impingement of energy is the chromaticity cone (CC). Using a family of concentric spheres, we describe a nonlinear transformation over the CC and construct a bijection from the CC onto the (flat) plane. In the process, we open up the CC and view it as a chart on the plane. Because the map is a bijection, the color perception information is preserved (invariant) through the transformation. We call this flat image of the CC, the “chromaticity chart”. We mark the isoX, iso-luminance (isoY) and isoZ lines and the geodesic lines on the chart. Further, we add infinity to the chromaticity chart and obtain a one point compactification of the entire chart plus infinity through a stereographic projection. The stereographic projection then lets us to view the entire compactified chromaticity chart plus infinity as an image on the unit sphere. In the XYZ space, ideally, arbitrary heights for the monochromatic energy are admissible, which means that the CC could be indefinitely long. By adding infinity to the chromaticity chart, the entire cone including the case of infinite energy gets mapped onto a compact sphere, the sphere’s pole corresponding to the infinite energy. This step maps the entire range for color perception due to spike type energy impingement, including the spikes of infinite heights, onto a compact unit sphere.