The tolerance unit opens every edge

A safety margin keeps a printed colour a stated distance inside what the press can print, and pushes every colour on the press’s boundary inward. How far depends on the corner it sits on. Nine black levels sharpened the press’s corners settled that for a margin measured in CIELAB. Read by their ink recipes, three of the census’s nine end colours are vertices of the press — the solid yellow, green and red, every ink at nought or full — and six sit along edges, one ink part-way. The recipe says in which directions each can move, the press model says what those directions are in CIELAB, and a perfect corner of that shape moves a predictable multiple of the margin. On a press sampled finely enough in black, every one of the nine meets its own shape within seven per cent.

A print buyer does not state a margin in CIELAB. Two ΔE₀₀ is the unit a specification uses, and a tolerance’s own unit rounds the corners repeated the census with depths and moves measured in ΔE₀₀. Almost every corner rounded, and the solid yellow stayed sharp. The yellow stays sharp because it leans explained the yellow by the direction of its tongue: a circular cone about an axis leaning towards lightness, rescaled by ΔE₀₀’s three unit lengths at the tip, came within ten degrees of the yellow’s opening. The same account failed at every other corner, and the reason given in the black-levels essay was that six of them are not cones.

A wedge seen through a different metric is still a wedge, with a different angle between its faces. ΔE₀₀ is locally a quadratic form — a small step’s ΔE₀₀ is the square root of the step against a symmetric matrix — and the matrix stretches some directions more than others. The prediction was that carrying each edge’s two faces into ΔE₀₀’s local coordinates at the end colour, and reading the new dihedral angle, accounts for all six edges within a tenth of the margin, because the CIELAB moves meet their wedges closely and ΔE₀₀ is smooth over a few units.

One rule for every corner #

Carried into ΔE₀₀’s local metric, each corner’s own shape predicts how far a ΔE₀₀ margin moves it on the finely sampled press: all six edges within a tenth of the margin at one, two and four ΔE₀₀, and all three vertices within a seventh. The six edges open from 67 to 141 degrees in CIELAB to 123 to 157 in ΔE₀₀ — nearly faces — so that their moves fall to 1.02 to 1.14 times the margin. The green and the red, vertices sharper than a cube’s corner in CIELAB, fall to about 1.2. Only the solid yellow’s cone stays sharper than a cube’s, at 2.04 against the census’s 1.91 to 1.95; the cone opens to 59 degrees where the margin found 62, and the lean account had said 52. The opening is ΔE₀₀’s chroma term: without it every corner would be as sharp in ΔE₀₀ as in CIELAB, or sharper.

• The prediction holds, and for the vertices as well as the edges. One construction — the recipe’s directions, the press model’s derivatives, ΔE₀₀’s local matrix — explains the whole ΔE₀₀ census.
• An edge in ΔE₀₀ is nearly a face. The census’s rounding in the tolerance’s unit is what a chroma compression does to a ridge pointing out in chroma.
• The yellow is sharp because it is very sharp. Its CIELAB cone is the narrowest of the nine, and a five-fold chroma compression opens it only to 59 degrees.
• The lean account is superseded, not contradicted. It reached the yellow’s opening by a different road and less closely.

The upper bar in each row is the black-levels essay’s result, the CIELAB shape. The middle bar is the same shape carried into ΔE₀₀, and the lower bar is the census — a ΔE₀₀ distance transform on the press, each end colour moved to the nearest cell two ΔE₀₀ deep. The middle and lower bars agree in every row. The upper bar, where every corner but the two half-black edges sits past a cube’s corner, is what a CIELAB margin costs; the two lower ones are what a tolerance margin costs, and apart from the yellow they are close to one.

A metric at a point #

ΔE₀₀ is not a distance in CIELAB, and at a single colour it is not even a distance in any fixed coordinates, because its weights depend on the pair’s mean lightness, chroma and hue. For small steps from one colour it behaves like one. The square of the ΔE₀₀ between a colour and a colour a small step away is, to second order, the step against a symmetric three-by-three matrix, and that matrix is read off the formula by differentiating it twice. Its square root is a map from CIELAB steps to a local coordinate system in which ΔE₀₀ is ordinary distance. Two uniform spaces disagree about between used the same matrix to ask which colours lie between two others, and found that ΔE₀₀ has one while CAM16-UCS’s published distance, a power below one, does not.

At the census’s saturated end colours a ΔE₀₀ unit along chroma is three to five CIELAB units, largest at the solid yellow, where the chroma is 91. Across hue it is one and a half to two and a half; along lightness one to one and a half, and one and a half only at the light yellows, where lightness is far from fifty. The chroma length is ΔE₀₀’s $S_C$, one plus 0.045 of the chroma: at chroma seventy a chroma difference is divided by more than four. These three lengths are nearly the whole matrix at these colours — the rotation term that couples chroma and hue acts only in the blues, and none of the census’s end colours is blue — so the local map is, to the digit, a stretch along three perpendicular directions.

Each edge opens #

An edge of the press is where two faces of the ink solid meet: for the yellow with three quarters of cyan, the face where yellow is at full strength and the face where magenta is at nought, with the free ink, cyan, running along the edge between them. In CIELAB the angle between those faces, measured across the edge, is 71 degrees. The ridge points out and a little down, into the chroma the two faces share.

Carried into ΔE₀₀, the direction out along chroma shrinks four-fold and the directions across it shrink less. A ridge pointing out in chroma is flattened: the faces swing apart, and the angle across the edge becomes 123 degrees. All four light edges do the same, from 67, 70, 71 and 92 degrees to between 123 and 129. The two half-black edges were already open in CIELAB, at 117 and 141 degrees, and open further, to 127 and 157. A wedge of dihedral angle θ moves its edge 1/sin(θ/2) times the margin, so the six edges’ predicted moves are 1.02 to 1.14 times a ΔE₀₀ margin. The census had them at 1.02 to 1.20 at one, two and four ΔE₀₀, each within seven hundredths of the margin of its own prediction.

That is the earlier census’s finding that a ΔE₀₀ margin rounds the corners, now with its mechanism. A tolerance has a grain found that one colour difference, pulled back into tristimulus values, is a long thin shape whose short axis points along a*; the edges here are opened by the same anisotropy, read around a corner instead of around a colour.

The vertices, and the one that stays sharp #

A vertex is a cone with several faces, one for each ink that can move off its limit: at the solid yellow, adding cyan, adding magenta, adding black and taking yellow away. In CIELAB the solid yellow’s cone is the sharpest corner in the census — a perfect cone of that shape moves 3.02 times a margin — and the green’s and the red’s move 2.11 and 2.30 times.

Carried into ΔE₀₀, the green’s and the red’s cones open to about 115 degrees, and their predicted moves fall to 1.18 and 1.17 — near the edges’, and the census has them at 1.20 to 1.25 and 1.06 to 1.19. The solid yellow’s cone opens only to 59 degrees, and its predicted move is 2.04, against the census’s 1.91, 1.95 and 1.91 at one, two and four ΔE₀₀. It is still sharper than a cube’s corner, the only one of the nine that is.

The yellow is the worst-predicted corner, and it misses by a seventh of the margin, over-predicting the sharpness by six or seven per cent. Its tip is where a tangent shape is most likely to overstate a corner: the cone’s faces curve most there, the press is thinnest, and a margin of one ΔE₀₀ is five CIELAB units along chroma — the scale at which a real tip starts to differ from its tangent. The edges, which the prediction was about, are within seven hundredths.

Why the yellow stays sharp #

The yellow’s persistence was the question the lean essay was written to answer, and the tangent cone answers it more simply. In CIELAB the yellow’s cone is far narrower than any other corner’s, opening only 39 degrees; the green and red open 56 and 51. A chroma compression of five opens every corner that points out in chroma, and it opens the yellow’s too, but from 39 degrees there is only so far to go. The others start wider and open past a cube’s.

The lean essay’s account reached the same corner by a different construction: it measured the tongue’s axis from the press’s cells — leaning towards lightness — fitted a circular cone about it, and rescaled that cone by ΔE₀₀’s three lengths. On the finely sampled press that account gives an opening of 52 degrees at a margin of two ΔE₀₀, against the 62 the margin found. The tangent cone gives 59. The lean was a real feature of the tongue, and the circular cone was the approximation that cost the missing seven degrees: the yellow’s cone has four faces, not a round cross-section, and its narrowest direction is not the one a circle about the lean axis assumes.

Which term does it #

The three lengths can be undone one at a time. Set the chroma length to one CIELAB unit, leaving hue and lightness as ΔE₀₀ has them, and every corner is as sharp as in CIELAB or sharper: the yellow moves 4.47 times the margin, the red 2.95, the green 2.63, and the light edges 1.36 to 1.93. Without its chroma compression, ΔE₀₀’s other two terms stretch the corners rather than open them.

Undo the hue length instead and the yellow rounds a little further, to 1.85; undo the lightness length and it sharpens, to 2.33. The other corners change by a few hundredths either way. ΔE₀₀’s hue term, as it stands at the yellow, sharpens it, and its lightness term rounds it, and the two nearly cancel. The lean essay found the hue function’s dip at yellow rounding the corner; that is the same term read against an average hue rather than against no hue term at all — the hue unit at yellow is shorter than at an average hue and longer than one CIELAB unit, and a longer one sharpens. The chroma term does the rest, and at every corner it is the rest.

What a specification should take from this #

A ΔE₀₀ margin costs a press almost nothing at its edges. A tolerance in the wrong coordinates is the general warning that the coordinates a tolerance is written in decide what it allows, and here they decide it in the buyer’s favour. An edge in ΔE₀₀ is nearly a face; a colour on one moves about 1.1 times the margin. That is the part of a margin costs a press its corners that a print buyer will meet, and it is much gentler than the CIELAB version.

The cost is at the light yellow corner. Only there does a ΔE₀₀ margin still push the corner in by twice itself, which in CIELAB is ten units along chroma for a two-ΔE₀₀ margin. A specification that holds every colour two ΔE₀₀ inside the press gives up the most saturated yellows first and almost nothing else at the corners.

The corner’s cost can be computed without the press’s volume. Everything here comes from the ink recipe, the press model’s derivatives at one colour, and ΔE₀₀’s matrix there. No distance transform is needed to know how far a margin will move a corner, and what one number accepts — the tolerance’s own shape — is the only other input.

How the corners were carried #

The end colours, their recipes and the finely sampled press are the black-levels essay’s: the coated press at a 320 per cent ink limit walked at thirty-three black levels, and the nine end colours of the gradient census. At each end colour ΔE₀₀’s quadratic form is its second derivatives by central differences at two hundredths of a CIELAB unit, and its symmetric square root is taken by a Jacobi eigendecomposition. Each direction the recipe can move in — one-sided differences of the press model at a hundredth of coverage — is mapped through the square root and normalised; an edge’s direction is the central difference along its free ink, mapped the same way. An edge’s dihedral angle is the angle between the outermost two mapped directions projected across the mapped edge; a vertex’s faces are the planes through pairs of mapped directions with every other on one side, and its move is the distance from the tip to the nearest point at unit depth from every face, found over every set of up to three active faces. The ablations rebuild the matrix from ΔE₀₀’s three unit lengths along the end colour’s lightness, chroma and hue, which reproduces the full matrix to the digit here, with one length set to one. The census is the ΔE₀₀ depth field of the tolerance-unit essay on the finely sampled press. With the identity for a matrix, every shape’s move is the black-levels essay’s CIELAB prediction to the last digit, which is checked.

What this leaves out #

The metric is taken at the tip. A margin of four ΔE₀₀ reaches twenty CIELAB units along chroma at the yellow, over which ΔE₀₀’s weights change. The predictions hold at four as well as one, but a margin of eight or ten would reach where they need not.

The corners are the census’s nine. They are all saturated, at chromas from 45 to 91, where the chroma compression is large. A corner of the press at low chroma — a dark three-ink neutral, a pale tint on paper — would open far less, and its ΔE₀₀ move would stay near its CIELAB one.

None of the end colours is blue. Near a hue of 275 degrees ΔE₀₀’s rotation term couples chroma and hue, the matrix is no longer three stretches, and a corner there could open or sharpen in ways the three-length ablation cannot separate.

Still open: the press’s blue corners #

The coated press has a blue corner of its own — cyan and magenta at full strength — and a violet edge running from it towards the magenta, and neither is in the census, because no census gradient ended there. It is the one region where ΔE₀₀’s rotation term acts, and a size is not a direction found that the angles a local metric puts between departures decide what they cost together — the angles a rotation term changes.

The calculation is the same construction at the blue vertex and along the blue-to-magenta edge: recipe directions, the press model’s derivatives, ΔE₀₀’s local matrix with its off-diagonal term, and the ΔE₀₀ margin census extended to those colours. The prediction is that the construction still accounts for them within a tenth of the margin, because it uses the full matrix, while the three-length ablation fails there by more than a tenth, because it leaves out the coupling. If the construction itself misses at the blue, ΔE₀₀’s local quadratic form changes too fast there for any tangent shape to describe a corner a few units across.

An explanation should be tried at the scale of the rule #

The habit is about what counts as having explained a set of cases.

The yellow’s sharpness in ΔE₀₀ had an explanation, particular to the yellow, that worked there and failed everywhere else. The explanation here uses nothing particular to any corner: the recipe’s directions, the press model’s derivatives, ΔE₀₀’s matrix. It covers all nine, and it covers the yellow more closely than the particular account did.

The failure mode is to explain the striking case on its own terms and leave the ordinary cases to a different story. The test is to ask what rule the striking case would follow if it were ordinary, and apply that rule to every case at once. Here the yellow turned out to be ordinary — just the sharpest corner, opened by the same compression as the rest.