Extended Planckian locus

The Planckian locus is a curve on a chromaticity diagram that records the color of a black body radiator for different temperatures. As temperature increases from 0 to ∞, red, orange, yellow, whitish, and bluish lights are generated, and these are broadly typical of the colors of everyday illuminations. The red for very low temperatures is on the edge of the spectral locus (it is monochromatic), but the bluest blue is in the middle of the chromaticity diagram, far from being a <i>pure</i> color. The Wien locus is parameterised by a simpler equation than the Planck locus and runs almost parallel to the Planckian locus. These two loci are so close together that a temperature conversion brings the corresponding chromaticities into an almost complete coincidence. However, the Wien locus is longer-extends more towards the short-wave part of the chromaticity diagram-than the Planck locus for an infinite color temperature. In this paper, we extend the <i>Planck</i> and <i>Wien</i> formulas to accommodate negative temperatures. The Planckian locus extends only slightly and stops in the middle of the chromaticity diagram. However, the Wien locus naturally extends all the way to intersect the spectral locus, at 360 nm. We show that the extended Wien locus is continuous: negative and positive infinite-color temperatures (the limit of the temperature as it tends to positive <i>and</i> negative ∞) converge to the same point. However, there is a substantial discontinuity at the limit of the temperature as it tends to positive <i>and</i> negative 0, evidenced by the large chromaticity difference between the violet and red ends of the Wien locus. This mathematical framework provides a firmer theoretical basis for widely used lighting indices such as correlated color temperature, thereby strengthening their practical applicability. Theoretical and practical results of this research are discussed.