Extended Wien and Planck Loci

Abstract

In colour science we are familiar with the Planckian locus, a curve in chromaticity space that is indexed by the temperature T of a blackbody radiator, that approximately delineates the colours of typical lights. In the CIE $(u',v')$ chromaticity diagram the reddest (lowest-T) light intersects the Spectral locus at 830 nm; as T rises, the lights become progressively orange, yellow, white and blue with the bluest light being defined by an infinite colour temperature. Interestingly the bluest blue Planckian chromaticity is far from the spectral locus. The Wien locus is parameterised by a simpler equation than Planck and runs almost parallel with the Planckian locus.

The loci are so close together that a temperature conversion brings corresponding chromaticities into a most complete coincidence. Though, the Wien locus is longer - extends more towards the short-wave part of the chromaticity diagram - than the Planck locus, for an infinite temperature.

In this paper, we extend the Wien formula to allow negative temperatures. When we do this the Wien locus naturally extends all the way to intersect the spectral locus (at 360 nm). The extension to negative temperatures allows the Wien locus to model any reasonable light’s chromaticity within its arc. We show that the extended Wien locus is continuous: negative and positive infinite colour temperatures converge to the same point. However, there is a substantial discontinuity at $T=0$ evidenced by the large $(u',v')$ difference between the blue and red ends of the Wien locus. A practical application of this result is discussed (via the theory of Locus Filters) to improve design of color-correction filters in photography.