A primary is chosen for four things

A display’s red primary is a point on the chromaticity diagram, and four different requirements have opinions about where it may be.

The claim #

No single requirement decides where a display’s primary may sit. Every primary’s boundary is held by at least two, in different directions — and one of the four holds no part of any boundary at all.

• The four are: how well a gain in the display’s own basis undoes a change of light; how much of the chromaticity diagram the three primaries enclose; how many real surfaces fall inside them; and whether a light of that colour exists at all.
• Realisability is a hard wall rather than a cost, and it holds forty-eight per cent of the red primary’s boundary and fifty per cent of the blue’s.
• Adaptation holds the rest of red’s and most of green’s, at forty-two and sixty-nine per cent.
• Surface coverage holds nothing anywhere. It is slack in every direction on all three primaries, by a wide margin.
• And the intersection of the four is a small fraction of the smallest of them — five per cent for red — because the regions cross rather than nest.

Why a primary has four requirements and not one #

The previous round drew one of these regions. A primary’s tolerance is a shape rather than a radius established that the set of positions a primary can take before the adaptation cost rises by one per cent is a long thin thing with a ratio of 15.1 between its longest and shortest directions, and that half of its boundary lies outside the region a real primary can occupy.

It also said what was missing: a display’s primaries are chosen for gamut, power, availability and stability, and adaptation is one term among them. Four of those are now drawn.

Each is written as a cost that is smaller when better and normalised to its value at the designed primaries, so a one per cent rise means the same thing in all four and no weighting has to be invented. Inventing one would be the ordinary thing to do and would make every number here a statement about the weights, which is the failure mode this collection has caught itself in twice.

The four #

Adaptation. A display’s basis is the inverse of its primary matrix, so the primaries decide the axes a von Kries gain is applied along — and how much of a change of light such a gain removes is a property of those axes. This is the requirement the previous round measured, and it is the one nobody outside this collection would think to put in a specification.

Gamut. The share of the spectral locus’s area the triangle of three primaries encloses. It is the requirement everybody knows about, it is what the gamut race is about, and it is the reason primaries have been pushed outward for three decades.

Surfaces. The share of this collection’s own reflectance family the display can actually show. It is a better-motivated version of the gamut requirement — an area on a diagram is not a count of colours anybody will need — and it is the requirement that turns out to be slack.

Realisability. Whether the chromaticity is inside the spectral locus at all. A point outside it is not a colour, so no emitter of any kind produces it: this is a hard wall and enters as a cost that is finite inside and infinite outside.

What holds which boundary #

The answer is different for each primary and is never one requirement.

• Red: realisability 48 per cent, adaptation 42, gamut 10, surfaces 0.
• Green: adaptation 69 per cent, gamut 31, the other two 0.
• Blue: realisability 50 per cent, adaptation 31, gamut 19, surfaces 0.

The map between the two planes has a shape of its own, and how far a primary can be moved in each is what makes the four requirements bind differently.

Red and blue are held mostly by the shape of the locus, which is not a design decision at all: both sit close to the spectral boundary, so half their directions run straight out of the space of colours. That is why half of red’s tolerance region and half of blue’s lie outside what a real primary can occupy — the previous round’s finding, now with a name for the constraint doing it.

Green is the free one. It is furthest from the locus in the directions that matter, so its boundary is decided by the two costs rather than by geometry, and its intersection retains seventy-four per cent of its smallest part against red’s five.

The shapes behind those shares are worth describing, because a share of a boundary is a summary and the pictures are not.

The adaptation region is long and thin, with a ratio of 15.1 on red, 6.6 on green and 5.9 on blue, and its long direction differs on each: 330 degrees on red, 135 on green, 240 on blue. The gamut region is long in the direction that keeps the triangle’s area — roughly along the line joining the primary to the opposite edge — and short across it, with ratios near ninety on every primary because area is nearly linear in one coordinate and nearly flat in the other.

Those two long directions are about thirty degrees apart on the red primary. That is what leaves the intersection at five per cent of the smaller of them, and it is the geometry the neighbouring essay is entirely about.

The requirement that never binds #

Surface coverage holds no part of any boundary on any primary, and the margin is not small: its region reaches the edge of the search in almost every direction on all three.

The reason is worth having because it is a fact about surfaces rather than about the method. Real reflectances are dull. The family this collection uses is built from measured-looking spectra with smooth shapes, and no surface can be as colourful as a narrowband source — the optimal-colour limit is far inside the spectral locus at every lightness, and real paints and dyes are far inside that.

So a triangle that covers a respectable share of the diagram covers essentially all of the surfaces, and moving one primary by the amount any other requirement permits does not change which surfaces are inside. The requirement is satisfied so comfortably that it cannot discriminate.

That is a useful thing to know and an uncomfortable one, because surface coverage is the requirement with the best claim to being what anybody actually wants. It is slack precisely because the industry has already won that argument: displays got wide enough for surfaces some time ago, and the gamut race since has been about the rest of the diagram.

The intersection, and what it leaves #

Putting all four together leaves a region that is a small fraction of any of its parts: five per cent of the smallest for red, sixteen for blue, seventy-four for green.

Green is the outlier for two reasons at once. Its adaptation region is the least elongated of the three, so it loses less to a crossing; and realisability does not reach it at all, because the designed green sits far enough inside the locus that a one per cent adaptation rise is reached long before the boundary of the space of colours.

So the three primaries of one display are three quite different design problems. Red and blue are constrained by where colours stop existing; green is constrained by the objectives. A specification that gave all three the same tolerance would be simultaneously loose on two of them and tight on the third, in the directions that matter.

What separates green is realisability, not roundness #

Green’s intersection is explained above by two things at once — that its adaptation region is the least elongated of the three, and that realisability does not reach it. The second is right and the first is not: blue’s adaptation region is the least elongated, at 5.9 against green’s 6.6, and blue keeps sixteen per cent where green keeps seventy-four.

Sorting the three by elongation and reading the other columns beside it shows which variable is doing the work:

The two primaries realisability binds keep 5 and 16 per cent; the one it does not keeps 74. Elongation runs the other way across the first two rows and cannot be the explanation. Red is worse than blue, and there the elongation is plausibly the reason — 15.1 against 5.9, with realisability holding about the same share of each — but between blue and green the wall is the whole story.

That is a cleaner result than the two-reason version. A primary near the spectral locus loses most of its room to the edge of colour space; a primary away from it keeps most of it, and how thin its objective regions happen to be is a second-order matter that only separates the two primaries the wall already reaches.

Adaptation holds more boundary than anything else #

Averaging each requirement’s share across the three primaries gives a summary the per-primary lists do not:

The requirement that appears in no display specification holds nearly half of the total boundary, and the requirement everybody argues about holds a fifth. That is a stronger statement than the red primary’s 42 per cent taken alone, and it is the one the closing history section is reaching for when it says the adaptation term is either a discovery or an artefact of taking this collection’s own objective seriously.

It is also the number to be most suspicious of, for exactly the reason that section gives. The adaptation cost is the only one of the four whose definition is this collection’s own; the other three are the locus, an area on a diagram, and a reflectance family. If the adaptation objective were replaced by a different but equally defensible one, the 47 would move and nothing else in the table would. Nothing here establishes that adaptation is the right second requirement — only that a second requirement of that kind holds more boundary than the first one anybody writes down.

The three long axes are not independent #

One more thing the region shapes carry that the boundary shares do not. The adaptation regions’ long directions are quoted as 330, 135 and 240 degrees for red, green and blue. A long axis has no head or tail, so reducing each modulo 180 gives 150, 135 and 60.

Red’s and green’s long axes are fifteen degrees apart; blue’s is ninety degrees from red’s and seventy-five from green’s. So two of the three adaptation regions are elongated in nearly the same direction on the diagram and the third is nearly perpendicular to both.

That is the pattern a two-axis account of adaptation would predict — red and green trading against each other along one opponent direction while blue trades along another — and it is worth flagging as a pattern rather than adopting as an explanation, because three regions is three points and a fifteen-degree agreement between two of them is not much evidence. What it does say is that the three primaries’ tolerance shapes are not independent objects: whatever sets the direction of red’s cheap axis very nearly sets green’s too, and a display maker moving both outward is moving them along nearly the same cheap direction rather than along two unrelated ones.

What was computed, and how #

Each region is a level set found by bisecting outward along forty-eight directions in the primary’s own chromaticity plane, with the other two primaries held. All four regions for a primary are swept along the same directions, which makes the intersection exact: every region is star-shaped about the designed primary, so the intersection’s radius along a direction is the smallest of theirs along it.

A direction in which a requirement does not reach a one per cent rise inside the search’s reach is reported as capped. Three of the four are capped in most directions, and reporting that rather than a number is what makes their slackness legible.

The realisability cost is a step rather than a smooth function, so its “level set” is the locus’s own boundary as seen from the primary. That is a legitimate region and a slightly different kind of object from the other three, and it is why it is described as a wall throughout rather than as a cost.

Where the model stops #

Four is not all of them. Power efficiency, the availability of an emitter with that chromaticity, stability over temperature and over life, and cost are all real and none is here. Every one of them would cut the intersection further, and at least two — efficiency and availability — would cut hardest in the same direction the gamut requirement pushes, since a saturated primary is usually a narrow one and a narrow one is usually dim.

And the primaries are moved one at a time. A display whose three primaries drift together is a different object with a tolerance in six dimensions, and the two-dimensional slices here are shadows of it rather than sections through it.

The one per cent budget is a choice, and the regions scale with it. What was checked is that the ordering — which requirement binds where — does not move between a tenth of a per cent and five per cent. The shares shift by a few points and no requirement changes rank.

The generalisation #

Two things generalise, and the second is the more useful.

The binding requirement is a function of direction, not a property of the design. Asking what limits how far this primary can move has no answer without a direction attached, and a specification that names one requirement per parameter has thrown that away. On this display the answer is the shape of the space of colours in half of red’s directions and the adaptation basis in most of the rest.

And a requirement that never binds is a sentence that can be deleted. Surface coverage is the requirement with the best motivation and the least discriminating power, and knowing that is worth as much as knowing which one binds: it says where not to spend effort, and it says that an argument about surface coverage is an argument about something already settled.

The contrast between that picture and the one at the top of this essay is the essay in two images. Same display, same four requirements, same one per cent budget — and one primary is boxed in by the edge of colour space while the other is limited by two objectives pulling in nearly the same direction. A tolerance is a property of a position, not of a device.

The pattern is the same one this phase found in a camera’s dyes, where the two requirements a physical argument nominated turned out to hold nothing and a colorimetric one held everything. A constraint’s mechanism being easy to state is unrelated to its being tight.

What a display maker would take from this #

The practical content is short and is different from what a gamut-first reading would give.

Red and blue have almost no room to move outward and a good deal inward. Their outward directions run into the locus, so the tolerance on those two primaries is asymmetric — a specification quoting ±0.004 in both directions is describing a symmetric box around a strongly asymmetric region.

Green has room in every direction and is bound by the objectives. Its tolerance is genuinely a matter of how much adaptation performance and coverage a maker is willing to give up, which means it is negotiable in a way the other two are not.

And a requirement about surfaces is not worth writing down. Whatever the display, if it covers a respectable share of the diagram it holds essentially all of the reflectances anybody photographs — so the argument for a wider gamut is an argument about lights, screens and saturated synthetic colours rather than about the world.

That last one is the least comfortable and the most useful. It says the gamut race has been, for some time, a race about a set of colours that does not include most of what a camera points at.

Who found it, and when #

The trade between gamut coverage and everything else is the oldest argument in display design and is usually posed as gamut against efficiency: a narrower primary is more saturated and passes less light, so a wider gamut costs brightness. That trade is real and is not in this essay.

What is unusual here is the adaptation requirement. That a display’s primaries decide the axes a chromatic adaptation transform runs along is a consequence of how colour management is implemented rather than a design goal anybody states, and it is not in any display specification. It holds forty-two per cent of the red primary’s boundary in this model, which is either a discovery or an artefact of taking one collection’s own objective seriously — and honestly, it is not clear which.

Where the ladder goes next #

Every region in this essay is drawn in chromaticity, because that is where the colorimetry lives and where a specification is written. Nobody manufactures in chromaticity.

What a maker of an emitter controls is a peak wavelength and a bandwidth, and carrying a tolerance region across that map changes its shape, changes which direction is cheap, and shows that most of a region drawn in chromaticity is a colour no single-peak emitter can produce at all.