A display's blue is sharp where the rotation is not

Keep every colour a stated distance inside what a device can reproduce, and the colours on the device’s boundary have to move inward. How far depends on the shape of the boundary where they sit: on a flat face a margin moves a colour exactly the margin, and at a sharp corner, where two or three faces meet at an acute angle, it moves it several times further. A margin costs a press its corners found a coated press’s corners paying up to three times the margin, and nine black levels sharpened the corners found how to predict each one from its tangent shape — the directions its recipe can move in, carried into CIELAB — within seven per cent.

The tolerance unit opens every edge carried the same shapes into ΔE₀₀’s local metric, the unit a print buyer states a tolerance in, and found ΔE₀₀ opening every corner towards a flat face. At each of the corners it read, ΔE₀₀’s local matrix was three independent stretches, along lightness, chroma and hue. It is not so everywhere. ΔE₀₀ has a rotation term, fitted in 2001 to the way ellipses of equal difference tilt in the blues, and near a hue of 275 degrees it couples a chroma step to a hue step — it shears the space as well as stretching it. The blue corner has nothing to twist went to the press’s blue corner to see what the shear did, and found it gripping hard, a correlation of −0.86 at one edge, and changing the predicted move by three hundredths: the press’s blue was blunt in CIELAB already, 1.25 times a margin, and nearly a face in ΔE₀₀.

That essay closed on the one place the shear might still find something to twist. A display does not mix inks; it adds lights, and its blue primary is a single narrow emission at full strength — which in CIELAB should be a spike rather than a gentle turn. The prediction was that a display’s blue corner is sharp in CIELAB, and that there, and only there, the rotation term changes the ΔE₀₀ prediction by more than a tenth of the margin: a sharp corner in the one place the formula shears.

Sharp in CIELAB, blunt in ΔE₀₀, and past the rotation #

A display’s blue corner is sharp in CIELAB, as predicted: a CIELAB margin moves sRGB’s blue 3.35 times the margin and BT.2020’s 3.17, more than two and a half times the press’s blue. Insetting the curved gamut solid directly finds sRGB’s blue moving 3.35, 3.34 and 3.32 times margins of one, two and four. But the rotation changes nothing worth the name. A display’s blue primary sits at a CIELAB hue of 306 degrees, thirty past the rotation’s centre, where ΔE₀₀’s matrix correlates chroma and hue by only −0.22; and ΔE₀₀ compresses the corner’s chroma so strongly that it moves only 1.42 and 1.26 times a ΔE₀₀ margin. The rotation changes those by 0.017 and 0.011. Along the edge from blue to cyan the hue falls through 275 degrees and the correlation reaches −0.87 — but there the edge is within five per cent of a flat face in ΔE₀₀. Across the six corners of both displays and every point along the three edges that meet at blue, the rotation never moves a prediction by four hundredths.

• The prediction is right about the sharpness. An additive blue primary is the sharpest corner of an sRGB display in CIELAB.
• It is wrong about where the blue is. The rotation was fitted to blues near 275 degrees; a display’s blue primary is violet-blue, at 306.
• It is wrong about what ΔE₀₀ does to a sharp corner. ΔE₀₀’s chroma compression, sevenfold at the primary’s chroma, blunts the spike before the rotation could twist it.
• Where the rotation grips, the corner is gone. On the cyan edge the shear is as strong as anywhere at the press, on a surface that ΔE₀₀ has nearly flattened.

Each pair of bars is one corner of one display, read in the two units. In CIELAB the corners run from 1.67 (sRGB’s cyan) to 4.39 (BT.2020’s green), and the blue is the sharpest of sRGB’s six. In ΔE₀₀ every corner of both displays lies between 1.18 and 1.78 — the whole spread CIELAB shows is compressed into a band barely half a margin wide. The blue, sharpest of sRGB’s corners in CIELAB, falls to the middle of the six in ΔE₀₀: its move is divided by 2.4, more than any other corner of sRGB’s.

Why an additive blue is a spike #

A display’s gamut is the image of a cube. Its linear drive signals run from nought to one on each channel, and every colour it shows is a point in that cube carried into XYZ by the display’s primary matrix and then into CIELAB. The blue corner is the drive (0, 0, 1). From it the gamut extends three ways — adding red, adding green, or reducing the blue — and the tangent shape is those three directions carried into CIELAB.

The blue primary is dark: sRGB’s has 7 per cent of the white’s luminance, L* 32, and a chroma of 134. Two of the three ways out of it head straight back down its chroma. Reducing the blue drops it towards black, losing chroma and a little lightness; adding green swings it towards cyan, losing chroma even faster and gaining lightness steeply, because green carries most of the luminance and CIELAB’s cube-root response is steep where the luminance is low. Seen from the third direction — adding red, towards magenta, which runs mostly across hue — those two are only 44 degrees apart on sRGB and 43 on BT.2020, one tilted a little down and one a little up from the chroma axis. The two faces that meet along the magenta edge therefore meet at a sharp angle, and the corner is a thin wedge standing out along the chroma axis, with its blade running towards the purples.

The edges say where the sharpness lives. The edge from blue towards magenta closes to 47 degrees in CIELAB on sRGB and 41 on BT.2020, far sharper than the edge towards black at 80 and 103 degrees or the edge towards cyan at 82 and 72. That one edge makes the corner a spike, and it stays sharp all the way along — 46 degrees at a drive of a half. It is the edge along which a display’s blue becomes its violet-magentas, and the reason a display can show saturated purples a press cannot — part of the volume neither gamut contains the other found sRGB holding outside the press.

Carried into ΔE₀₀ that edge opens to 117 degrees on sRGB and 121 on BT.2020, and on sRGB the edges towards black and cyan open to 89 and 122. The mechanism is the one the tolerance unit opens every edge found at the press: ΔE₀₀ divides a chroma step by 1 + 0.045C, which at the blue primary’s chroma of 134 is about seven, while it leaves a lightness step nearly alone. Both faces of the magenta edge run mostly along chroma; divided by seven in that direction, the wedge between them opens towards a face. A spike in chroma is exactly the kind of corner ΔE₀₀ was built to flatten, because its data said observers are least sensitive to chroma differences where chroma is greatest.

Where the rotation is, and where the blue is #

ΔE₀₀’s rotation term is a product of two factors. One grows with chroma and is nearly one for any chroma above about 40. The other is a bell centred on a hue of 275 degrees with a width of 25, and at 306 degrees it has fallen to a fifth of its peak. Both displays’ blue primaries sit at 306 degrees, and there the rotation correlates a chroma step with a hue step by −0.22 and −0.23. The press’s blue was at 287, near the bell’s top; a display’s is out on its shoulder, because an additive blue primary is chosen as short a wavelength as a phosphor or filter can make bright, and short-wave blues look violet.

Walking away from the blue changes that in one direction only. Towards magenta the hue rises past 310 and the correlation fades towards nought. Towards cyan the hue falls, through 290 at a drive of a tenth to a seventh and through 275 at a drive of a fifth to three tenths, and the correlation reaches −0.87 on both displays — as strong as anywhere on the press. If the rotation is to act on a display at all, it acts on the edge between blue and cyan, on the bluest of a display’s sky colours.

It does not act there either. At the point on the blue–cyan edge where the correlation peaks, the edge’s two faces meet at 144 degrees in ΔE₀₀ on sRGB and 155 on BT.2020 — the edge moves 1.05 and 1.03 times a ΔE₀₀ margin, within five per cent of a face. The rotation changes those by −0.035 and −0.034. The largest difference the rotation makes anywhere near either display’s blue is 0.038, on sRGB’s cyan edge at a drive of a twentieth, where the hue is still 301 and the correlation only −0.34, but the edge is still sharp enough for a small shear to show. Where the shear is strong the edge is flat; where the edge is sharp the shear is weak. On a display, as on the press, the two never meet.

Insetting the solid itself #

Everything so far is a prediction from the tangent shape: a perfect cone or wedge with the corner’s own faces, and the distance a margin moves its apex. A display’s gamut is not a perfect cone — its faces are images of the cube’s faces under CIELAB’s cube root, and they curve — and a sharp corner is where curvature could matter most, because a thin spike measured a few units in is a spike measured where its faces have already bent. The earlier essays checked their predictions against a finely sampled census of the press; here the gamut is analytic, so the check can be exact.

For each display, margin and unit, the census finds the nearest colour to the blue corner that lies at least the margin inside the curved solid, searching every direction into the solid for the first point at the margin’s depth, measured against the boundary sampled densely near the corner. In CIELAB sRGB’s blue moves 3.35, 3.34 and 3.32 times margins of one, two and four, against the tangent shape’s 3.353. In ΔE₀₀ it moves 1.416, 1.409 and 1.395 at one, two and four, against 1.421. BT.2020’s blue moves 3.168, 3.171 and 3.169 in CIELAB against 3.173, and 1.261, 1.260 and 1.255 in ΔE₀₀ against 1.264. The faces curve, but not fast enough to matter over four units: the tangent shape is the corner, to about a per cent. That settles the construction at the one corner where a spike might have outrun it.

What a specification should take from this #

A ΔE₀₀ margin on a display costs about the same at every corner. In CIELAB a margin costs a display’s corners anything from 1.7 to 4.4 times itself, and the blue or the green is the corner that pays most. In ΔE₀₀ the six corners of both displays pay between 1.18 and 1.78, and the blue pays 1.42 and 1.26. A specification written in ΔE₀₀ that allows the same margin at every hue is, on a display, close to allowing the same cost at every corner — which a CIELAB specification is not. A tolerance unit rounds the corners found the same levelling on the press; a display, whose corners are far sharper in CIELAB, is levelled further.

The rotation term is a statement about small differences near a hue, and nothing more. It exists because MacAdam’s ellipses tilt in the blues, and the tilt changes which of two nearby colours looks more different. It does not change the shape of a gamut enough to move a margin’s cost, at a press or at a display. Where the formula is not smooth found ΔE₀₀’s blue region carrying its discontinuities; its rotation, by contrast, is smooth and small where it meets a boundary.

Know which corner a hue belongs to. “The blue corner” names a drive, and on a display that drive is at 306 degrees, while the blues ΔE₀₀ was fitted on are at 275, on the edge between the blue and cyan primaries. A tolerance or a gamut-mapping rule written for “blue” and tested at the display’s blue primary has been tested somewhere the formula’s blue behaviour is not.

How the corners were read #

Each display’s gamut is the image of the unit cube of linear drive under its own primary matrix — sRGB’s and BT.2020’s, both with D65 white — carried into CIELAB against the display’s white. A corner’s tangent shape is the three directions its drive can move in, taken as finite differences of a ten-thousandth of drive; an edge’s is its two outward directions and its own direction along the free channel. Carried into ΔE₀₀, each direction is mapped by the symmetric square root of ΔE₀₀’s local quadratic form at the corner, built from second differences of ΔE₀₀ itself; without the rotation, by the matrix rebuilt from ΔE₀₀’s own lengths along lightness, chroma and hue with no coupling between them. A pointed cone’s move per unit of margin is the shortest vector whose component along every face’s inward normal is at least one; a wedge’s is one over the sine of half its dihedral. The rotation’s grip is the correlation the local matrix puts between a unit chroma step and a unit hue step.

The direct inset samples every face of the drive cube near the blue corner on a grid crowding towards the edges, keeps the samples within 45 CIELAB units, and finds, along each of 1,600 directions into the solid, the first point whose distance to the sampled boundary reaches the margin — by bisection, with CIELAB distance or ΔE₀₀ — then refines about the best direction with a seeded search. Its depth is then checked against the faces themselves by direct minimisation, and lands at the margin within two per cent.

What the census leaves out #

Two displays, both with D65 white. A display calibrated to another white has other corners in CIELAB, and a display whose blue primary lies at another wavelength puts its blue at another hue — BT.2020’s, far more saturated than sRGB’s, lands at 306 degrees too, but a different primary need not.

CIELAB against the display’s own white. A gamut read against a different adapting white — a print’s paper, or a room’s light — has corners of other shapes. The earlier essays on this press read it against its paper, and this one reads each display against its own white; a comparison between them compares two adaptations as well as two devices.

The margins are small. One to four units is where a tangent shape is meant to hold, and it holds. A margin of twenty would reach where the faces have bent, and a spike measured that far in is no longer the spike at its tip.

Still open: why the yellow is a display’s sharpest corner in ΔE₀₀ #

The ranking of a display’s corners turns over between the two units. In CIELAB sRGB’s blue is the sharpest and its yellow among the bluntest, at 2.41; in ΔE₀₀ the yellow is the sharpest of all six on both displays, at 1.75 and 1.78, while the blue has fallen to 1.42 and 1.26.

The calculation is this census at the yellow corners, their tangent shapes and edges in CIELAB and ΔE₀₀, with ΔE₀₀’s three lengths — along lightness, chroma and hue — read off at each, and the corner inset directly as here. The prediction is that the yellow keeps its sharpness because its faces run along lightness rather than chroma: a display’s yellow sits at L* 97, where ΔE₀₀’s lightness length is at its largest and compresses a lightness step by about 1.7, while its chroma, 97, is compressed by 5.4 — so the edge running down from yellow towards green and red opens less than any other. If it does, the yellow is where a ΔE₀₀ margin costs a display most, and it is the lightness weighting, not the chroma one, that decides it.

A strong cause needs something to act on #

The press’s blue corner showed a strong rotation acting on a blunt corner and changing nothing. The display was meant to supply the missing sharpness, and did — at the wrong hue, and in a unit that took the sharpness away before the rotation could reach it.

The failure mode is to look for an effect where each of its causes is largest separately, and assume they are largest together. The shear is largest at 275 degrees; the corner is sharpest at the primary; the two are thirty degrees apart, and between them ΔE₀₀’s own chroma compression flattens whatever the shear would have twisted. The test is to find the place where every factor is present at once — and here there is none.