Colour Science & Perception
what the eye settles, and what it cannot
Concept

Beer lambert — where it appears

The rule that a medium's absorbance is proportional to the concentration of what absorbs and to the path through it, so transmittance falls exponentially in both. It makes a colorant's depth a multiplier on absorbance rather than on colour, which is why doubling a loading squares a transmittance and steepens every edge in the spectrum.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

The same medium at six path lengths. Beer's law at 6 depths of one absorbing medium. The absorption coefficient is a single spectrum and the only thing changing is how far the light travelled, yet the patches differ in hue by 7.8° as well as in lightness — because absorption is exponential in depth and the observer is linear, so the bands that survive at d = 8 are not a scaled copy of the ones that survive at d = 0.25. Path length belongs to the geometry, not to the substance, which is why this sits in a field about scenes.

The colour is in the thickness

Transmittance is exponential in path length and the observer is linear, so doubling the depth of an absorbing medium squares the transmittance rather than halving the colour. A translucent object therefore has no one colour — its thin edge and its thick middle are different spectra of the same substance, and the hue moves between them.

scene · Scene How steep an edge a band draws, and what it costs. A Gaussian absorption band of stated width produces a reflectance edge whose own width depends on how deep the band is, because the exponential saturates: where the absorbance is large the reflectance is already nought and the edge is over. A forty-nanometre band at an absorbance of 3 draws an edge 32 nanometres wide; at 12 it draws one 21 nanometres wide. Below an absorbance of 2.3 the band never reaches a reflectance of a tenth at all and has no edge in this sense. The dashed lines are each band's own width, which is the number a slope limit would have been given.

A sharp edge is bought with depth

A slope limit on reflectance was introduced as the weakest honest statement of a pigment's bluntness, with a band-shape limit named as the stronger version to be written later. Written, it is not stronger. Three absorption bands none narrower than forty nanometres reach further than a forty-nanometre slope limit in 94 of 154 directions of the object-colour solid, because a band's width and the width of the reflectance edge it draws are different quantities — and what converts one into the other is how much colorant is in the film.

limits · Limits The same lamps, charged by two kinds of slow pigment. Each lamp — the three emitters, a fourth band at seven positions, and the cheapest symmetric and lopsided bands rendering at 88 — as its fidelity index against the directions of the solid a slow pigment loses more than a per cent of its reach in, for a forty-nanometre slope limit (open) and for three absorption bands none narrower than forty nanometres (filled), joined by a line. Every line runs leftward: the band-shape pigment loses fewer directions under every lamp, and the lamps rendering best gain the most. At 88 the slope limit loses 50 and the band-shape pigment 33.

The pigment decides the trade, not the band

A four-emitter lamp that renders well costs a slow pigment part of its reach, and reshaping the fourth band barely changes that. Changing the pigment changes it a great deal. Modelled as three absorption bands none narrower than forty nanometres, rather than as a forty-nanometre slope limit, a slow pigment loses a third fewer of the solid's directions under the best-rendering lamp, and the price of going from the three-emitter lamp to a fidelity of 88 halves. The lopsided band that helped a slope-limited pigment does nothing at all for this one. And at 590 nanometres a fourth band renders at 86 almost for free.

limits · Limits Every lamp, charged by a slope limit and by bands in nanometres and in energy. For each lamp, ordered by fidelity index, the directions of the solid lost by a forty-nanometre slope limit, by three absorption bands none narrower than forty nanometres, and by three none narrower than 0.158 electronvolts. The energy convention loses fewest under every lamp: 20 against 23 and 31 under the three emitters, and 26 against 33 and 51 under the best-rendering lamp.

Written in energy, the bands spare the green

A slow pigment modelled as absorption bands pays half what a slope limit pays for a four-emitter lamp's rendering, and a fourth band at 590 nanometres costs it almost nothing. That result assumed every band is at least forty nanometres wide, and a real absorption band has a width in energy, not in nanometres. Written that way it is narrower in the blue and wider in the red, and the fear was that the red would take back the amber. It does not. The band limit written in energy loses fewer directions under every lamp, the fourth band at 590 still costs six, and the price of rendering at 88 falls from ten directions to six — because the directions it wins back are green, not amber.

limits · Limits

Named alongside it

The objects these essays reach for when they reach for this one.

AbsorptionObject-colour solidOptimal coloursPigmentColour renderingModelling assumptionNarrow band displaysReflectanceSpectral power distributionBoundChromaticityCIELAB

All concepts