The blue corner has nothing to twist

Keep every printed colour a stated distance inside what a press can print, and each colour on the press’s boundary has to move inward. How far depends on the corner it sits on, and two essays have now settled how to predict it. Nine black levels sharpened the press’s corners read each corner’s shape from its ink recipe — the directions the recipe can move in, which the press model turns into CIELAB — and found every one of the margin census’s nine corners moving within seven per cent of what a perfect corner of that shape would, once the press was sampled finely enough in black. The tolerance unit opens every edge carried the same shapes into ΔE₀₀’s local metric and predicted the census in the unit a print buyer states: edges within a tenth of the margin, vertices within a seventh.

Both essays had the same nine corners, and none of them is blue. At each of them ΔE₀₀’s local matrix — the quadratic form that turns a small CIELAB step into a colour difference — was three independent stretches, along lightness, chroma and hue, and rebuilding it from those three lengths alone gave the same predictions to the digit. That is not true everywhere. ΔE₀₀ has a rotation term, added in 2001 to fit 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. A matrix with a large coupling shears the space: it does not just stretch along chroma and hue, it turns one into the other.

The tolerance-unit essay closed on the press’s blue corner — cyan and magenta at full strength — and the edges beside it, the one region where the coupling acts. The prediction was that the construction with the full matrix still accounts for them within a tenth of the margin, while the three-length rebuild, which leaves the rotation out, misses by more than a tenth. If the construction itself missed at the blue, ΔE₀₀’s matrix would be changing too fast there for a tangent shape to describe a corner a few units across.

The rotation is strong and has nothing to act on #

At the press’s blue corner ΔE₀₀’s local matrix correlates a chroma step with a hue step by −0.73, and at the edge with three quarters of magenta by −0.86 — where every one of the census’s nine corners had nought to three decimals. But the blue corner and its edges are blunt in CIELAB already: their tangent shapes move 1.22 to 1.27 times a margin, where the census’s corners moved up to 3.02. Carried into ΔE₀₀, every one of them is within a tenth of a flat face, and the construction predicts the ΔE₀₀ census within a tenth at margins of two and four. Without the rotation term the prediction changes by at most 0.035. The finely sampled press matters here too: on the census’s nine black levels the blue corner moved 1.68 times a four-unit margin, a third more than its shape; on thirty-three it moves 1.24.

• The construction holds at the blue, as predicted.
• The rotation term does not matter, against the prediction: a shear of a nearly flat corner is a shear of nearly nothing.
• The press is blunt where ΔE₀₀ twists. Its sharp corners are its light yellow, green and red; its blue is a gentle turn.
• The black sampling had sharpened the blue as well, by the same notches it had cut in every other corner.

Each row is one colour, and its four bars read the same corner four ways. The first two are CIELAB: the shape’s prediction and the census on the finely sampled press. The last two are ΔE₀₀: the shape carried by the full matrix, and the census. Every bar sits between one and 1.54, and most between one and 1.3. The census’s yellow, in the same figure for the earlier essays, ran to three.

Six colours between cyan and magenta #

The census’s colours came from the ends of ten gradients chosen because their straight lines left the press, and the ring they were drawn from has colours all the way round. Six sit at the blue. The blue corner is cyan and magenta at full strength and nothing else: a vertex, every ink at nought or full. Four edges lie either side of it: full cyan with magenta at a half or three quarters, and full magenta with cyan at a half or three quarters — one ink part-way, free to slide along the edge. And the blue corner with half black is an edge too, black being the free ink. They span 262 to 320 degrees of CIELAB hue and chromas of 30 to 49.

For each, everything the two earlier essays computed for the census’s corners: its tangent shape from the press model’s derivatives — a wedge for an edge, a pointed cone for the vertex — and the move a perfect corner of that shape makes per unit of margin; its move under CIELAB margins of one to eight units on the press walked at nine and at thirty-three levels of black; the same shape carried into ΔE₀₀ by the full local matrix and by the three-length rebuild; and its move under ΔE₀₀ margins of one, two and four on the finer press.

Where ΔE₀₀ couples chroma and hue #

The rotation term is ΔE₀₀’s answer to the MacAdam-style ellipses that lean in the blue: the formula’s ellipses of equal difference, drawn in the a*b* plane, are tilted there and nowhere else. Its size is a product of two factors — one that grows with chroma and saturates by about 40, and one that is a Gaussian in hue centred on 275 degrees, thirty degrees at its peak and falling to a third of that 25 degrees either side. At every one of the census’s nine corners, the correlation it produces between chroma and hue steps rounds to nought. At the blue it is the largest term in the matrix: −0.71 at 262 degrees, −0.86 at 277, −0.73 at the blue corner itself at 287, −0.33 at 301 and −0.04 at 320, as the Gaussian falls away towards magenta.

A correlation of −0.86 is not a small effect on a colour difference. It means a step that increases chroma and turns the hue one way counts as much less than the same step with the hue turned the other way — the whole of the blue ellipses’ tilt. If a corner’s move were sensitive to the matrix’s shape, this is where it would show.

The blue is blunt in CIELAB #

The census’s corners were sharp: the solid yellow a cone moving 3.02 times a margin, the red and green 2.30 and 2.11, the light cyan-and-yellow edges wedges of 67 to 92 degrees. The blue corner’s cone moves 1.25 times a margin, and its four edges’ wedges open 104 to 111 degrees, moving 1.22 to 1.27; the half-black blue edge opens 138 degrees and moves 1.07. On a press, cyan and magenta overprint to a blue that is dark and not very saturated, and the faces meeting there — cyan and magenta each at their limits, and black’s first increments darkening the corner — meet at wide angles.

On the finely sampled press the blue corner and its edges meet their shapes as the census’s corners did: within an eighth at margins of four and eight units, and all but the half-black edge within seven per cent. On the census’s press of nine black levels they did not. The blue corner moved 1.68 times a four-unit margin there, 34 per cent beyond its shape, and its edges 1.44 to 1.70 — the same notches between black levels that sharpened the yellow and the green, cut into the blue as well, where black is one of the directions every one of these recipes can move in.

The cross-section shows the notch directly. Through the edge of full cyan with three quarters of magenta, four units out, the coarse press subtends 104 degrees and the fine one 125, against the 107 degrees the edge’s faces make in the press model. The coarse press is narrowest close in — 82 degrees two units out — where the notch between black levels sits; the fine press is wider than the faces’ angle within four units, and the two agree from six units out. A margin of a few units reads the region where they differ, and the difference between the two samplings is the difference between a corner that moves 1.44 times a four-unit margin and one that moves 1.30.

With the rotation and without it #

Carried into ΔE₀₀ by the full matrix, the six shapes predict moves of 1.02 to 1.05 times a ΔE₀₀ margin; by the three-length rebuild, 1.02 to 1.07. The largest difference between the two is 0.035, at the edge with three quarters of magenta, where the coupling is strongest. The census, on the finely sampled press, has the six at 0.98 to 1.06 at margins of two and four ΔE₀₀, every one within a tenth of the full matrix’s prediction — the tolerance-unit essay’s result, holding at the blue.

The reason the rotation changes so little is geometric. A margin’s move at a corner depends on how sharply the corner’s faces meet; a shear changes the angles between directions, and it changes a sharp angle by a lot and a nearly straight one by little. Carried by ΔE₀₀’s chroma compression, the blue edges open to 145 to 156 degrees — within a quarter of a straight line — and a shear that turns chroma into hue by three quarters can open or close a 150-degree wedge by only a few degrees either way. At 1/sin(θ/2), a few degrees near 150 is a few hundredths.

The prediction reasoned from the size of the coupling. What decides the effect is the size of the coupling times the sharpness of the thing it acts on, and at the blue the second factor is near nought.

Why the blue is where formulas disagree #

The blue has been the awkward region of every colour-difference formula examined here, and the reason is the data rather than the formulas. MacAdam measured it: his ellipses of just-noticeable difference are smallest and most tilted in the blues, and every space since has had to bend to fit them. ΔE₀₀ bends with its rotation term, and where the formula is not smooth found that the bend has a hard edge in it — a discontinuity in the hue term that sits where many industrial samples live. A size is not a direction found the local metric’s angles deciding what two departures cost together, and the rotation term is the one part of ΔE₀₀ that changes those angles rather than the lengths.

So the prediction had good reasons to expect the blue to be where the construction strains: a formula term fitted to the region’s peculiar data, strong there, and shaped to change angles. None of that reaches a margin’s cost at the press’s blue corner, because the press has no sharp corner there for an angle-changing term to act on. A tolerance is a shape, and at the blue the tolerance’s shape is tilted; the press’s shape there is nearly flat, and a tilted ball rolled against a flat wall touches it the same way whichever way it is tilted.

That also bounds what the rotation term could do anywhere on the press. Its effect on a corner is at most a few degrees of angle, and a few degrees matter only to a corner already sharp. The press’s sharp corners — the light yellow, the green, the red — lie where the rotation term is zero.

What a specification should take from this #

A ΔE₀₀ margin costs the blue nothing. Every colour at the press’s blue moves about 1.03 times a ΔE₀₀ margin, as a colour on a flat face would. The rotation term that makes ΔE₀₀’s blue ellipses lean does not make a blue margin cost more or less.

The costly corner is still the light yellow. A tolerance’s own unit rounds the corners and the tolerance-unit essay found the solid yellow the one corner a ΔE₀₀ margin still pushes in by twice itself; the blue is at the other end of that range.

The construction is general. Nine corners, then six more in the one region the formula behaves differently, and the same three inputs — the ink recipe, the press model’s derivatives, ΔE₀₀’s local matrix — predict every one. What one number accepts is the tolerance’s shape; the corner’s shape is the press’s; the margin’s cost is the two together.

How the corners were read #

The colours are those of the press’s hue ring at the blue, printed from their recipes by the site’s coated press model at a 320 per cent ink limit. Their tangent shapes, their CIELAB moves on the press walked at nine and thirty-three black levels, and their ΔE₀₀ moves are computed exactly as the two earlier essays computed them for the census’s corners: one-sided differences of the press model at a hundredth of coverage for the directions; the dihedral angle across an edge, or the vertex’s cone and the nearest point at unit depth from its faces; each colour moved to the nearest cell centre at least the margin deep, by a Euclidean or a ΔE₀₀ distance transform of the voxel press. ΔE₀₀’s local matrix is its second derivatives at the colour; the three-length rebuild takes its unit lengths along the colour’s lightness, chroma and hue as three perpendicular stretches. The coupling is the matrix’s chroma–hue entry over the square root of its chroma and hue diagonal entries.

What this leaves out #

The blue is one corner of the press. The coupling acts over about fifty degrees of hue, and at chromas above about forty; the press’s blues are near the bottom of that chroma range, and a wider-gamut device with a saturated blue corner — a display’s blue primary, a violet ink — would put a sharp corner where the rotation is strong.

The margins are small. One to four ΔE₀₀ reach up to about twelve CIELAB units along chroma at the blue, over which the rotation term’s hue factor changes appreciably; the tangent matrix is taken at the corner.

The press is a model. Its blue is dark and dull because its cyan and magenta inks overprint that way; a press with a purer blue would have a sharper blue corner.

Still open: a display’s blue corner #

The rotation term was fitted to data around 275 degrees at chromas far beyond a press’s blues, and a display reaches them. The calculation is this construction at the blue corner of a display’s gamut — sRGB’s and BT.2020’s, whose blue primaries are corners in CIELAB of a very different sharpness from the press’s — with the margin census taken on the display’s solid in both units. The prediction is that a display’s blue corner is sharp in CIELAB, because an additive primary is a spike, and that there, and only there, the rotation term changes the ΔE₀₀ prediction by more than a tenth: a sharp corner in the one place the formula shears. If it does not, the rotation term is a correction to how ΔE₀₀ measures small differences that never reaches how a margin moves a gamut’s corners.

Strength of a cause is not size of an effect #

The habit is about predicting an effect from one factor of a product.

The rotation term is the most anisotropic thing ΔE₀₀ does, and the blue is where it does it; the prediction followed that to a large effect at the blue corner. The effect of a shear on a corner is the shear times how sharp the corner is, and the press’s blue corner is nearly flat. The failure mode is to rank causes by their own size when what they act on is small. The test is to find the other factor before predicting from the first — here, reading the corner’s shape before reading the formula’s coupling.