A size is not a direction

An audit that finds six independent things wrong with a calculation is immediately asked what all six cost together, and the arithmetic everybody reaches for is the one for independent errors: square them, add, take the root. That arithmetic requires the six to be orthogonal, and nothing in the audit ever asked whether they were.

The claim #

Two colour differences do not compose from their two sizes, and the quantity that completes them is an angle that is computable from measurements already taken and has never been computed.

• The angle between two of the audit’s departures runs from 1.3° to 179.4°, with a median of 94°, over the two thousand five hundred and twenty pairs available.
• On 1,095 of those the two together cost less than the larger of them alone, because the deviations partly cancel.
• The cosine rule in the local metric predicts the composition to 1.2 per cent at the median and 3.6 per cent at the ninth decile.
• Pythagoras is out by 28 per cent at the median and 77 at the ninth decile, and it is the rule an audit’s reader has no alternative to.

Where the arithmetic comes from #

Adding errors in quadrature is not a convention; it is a theorem, and the theorem has a hypothesis. If two deviations are vectors and the thing measuring them is a Euclidean length, then the length of the sum is the root of the sum of squares exactly when the two are perpendicular, and otherwise it is the cosine rule.

Colorimetry supplies the vectors, and supplies them exactly. A deviation is a tristimulus difference and two of them add exactly, because the colour integral is linear in the stimulus and in the observer’s curves alike. That half is not in doubt and it is the half the previous round rested its identities on.

What colorimetry does not supply is the length. ΔE₀₀ is not a Euclidean length in tristimulus values, or in CIELAB, or in anything: it is a compression, then a chroma, then a hue angle, then three weighting functions and a rotation term. Locally — for deviations small against the curvature — it behaves like a quadratic form, which is the shape this collection has measured elsewhere as a bowl, and a quadratic form has an inner product in it, so there is an angle. Globally it does not behave like anything with a closed form.

So the missing quantity exists, is well defined, and is a number about a pair of deviations rather than about either one. An audit reporting only sizes has thrown it away, and cannot get it back from the table.

The round did ask an interaction question and the question it asked was a different one. Its table takes two audits — the observer and the wavelength grid — and reads their combined effect under a series of lights, finding that on a light with a feature narrower than the grid step the two do not compound at all. That is a real finding about a mechanism, and it is the one the grid and the observer share. It is not the arithmetic of composition, and it does not scale to the fifteen pairs inside the observer audit alone.

What the angle is #

The local metric is the quadratic form that reproduces ΔE₀₀ for small deviations around one colour. Its entries are read off the formula rather than differentiated symbolically, because ΔE₀₀’s hue-rotation term is piecewise and the point of building the form is to predict compositions that the formula itself then checks.

Given the form, the angle between two deviations is the ordinary one: the inner product divided by the two lengths. It is dimensionless, it depends on the colour the two departures land on, and it is available the moment both deviations are known — which is the moment the audit computed either of its two numbers.

The distribution is not concentrated anywhere and in particular not at ninety degrees. Its median is 94° over the widened surface set, which is close enough to a right angle to be mistaken for support for the quadrature rule and is nothing of the kind: a median near ninety with a spread from one to a hundred and seventy-nine is a distribution about which the mean is uninformative.

Four hundred and forty of the two thousand five hundred and twenty pairs sit under thirty degrees, where the two deviations point nearly the same way and their costs very nearly add. Six hundred and fifteen sit above a hundred and fifty, where they point nearly opposite and largely cancel. Between them, a broad middle where neither rule is right.

There is a second reason the angle is not guessable, and it is the one that makes the whole exercise necessary rather than merely tidy. The angle is a property of three things: the two departures and the colour. Fix the two departures and walk the colour across the space, and the angle moves through most of its range.

The lens against the macular pigment runs from one degree on one surface to a hundred and fifty-six on another, over the same family. The same two eyes, disagreeing about two different samples, are reinforcing on one and cancelling on the other. There is no summary of that pair which is a number, and the audit’s table has one entry for each of them and none for the pair.

What the two rules predict #

The test is direct. For each pair on each surface, add the two tristimulus deviations, price the sum once, and compare against what each rule says it should have been.

The cosine rule is out by 1.2 per cent at the median and 3.6 per cent at the ninth decile. That residual is real and it is the curvature: the local form is a linearisation and the departures are not infinitesimal, so a second-order rule leaves a third-order error. It is small enough that the composition can be treated as solved.

Pythagoras is out by 28 per cent at the median and 77 at the ninth decile. On the audit’s own forty-two surfaces, where the range of colours is narrower, the figures are 1.7 and 26 — the same story with less spread in it.

The gain is a factor of more than twenty in the median error, from one extra number per pair. And the number costs nothing to compute: both deviations were already in hand, and the local form is nine numbers evaluated by six calls to the difference formula the audit was already calling.

The pairs that cancel #

The consequence a reader would most want is not the improved accuracy. It is that some pairs make things better.

On 1,095 pairs of 2,520 — forty-three per cent — the two departures together cost less than the larger of the two alone. That is not a small correction to a conservative estimate; it is the estimate having the wrong sign of error on nearly half its cases.

The mechanism is visible once the deviations are drawn rather than tabulated. A lens that has yellowed and a macular pigment that is dense are both filters that absorb in the short wavelengths, so on many surfaces they move the reading in nearly the same direction and their costs add. A rod intrusion adds a fourth curve into all three channels, which moves the reading mostly along lightness, and a cone optical density broadens rather than raises, which moves it mostly along chroma. Two departures moving a reading along different axes of a metric whose axes have wildly different lengths can land almost anywhere.

An observer who has two of the audit’s departures at once is not, in general, worse off than an observer who has the larger of them. The audit’s arithmetic says they are, and says it about nearly half the pairs it could have been asked about.

The four regimes, and how to tell them apart #

The angle is a continuum and the practical consequences fall into four regimes, which is the form the finding takes when a specification has to act on it.

Below thirty degrees the two departures reinforce and their costs very nearly add. Four hundred and forty of the pairs are here. This is the case a reader assumes never happens and it is the dangerous one, because an audit that added its terms in quadrature has under-stated the total by up to the difference between adding and rooting a sum of squares — forty-one per cent for two equal terms.

Between thirty and ninety the two reinforce partly. Quadrature is conservative here and the error is modest.

Between ninety and a hundred and fifty they partly oppose. Quadrature over-states, by up to the same forty-one per cent, and the total is smaller than the larger of the two on a growing fraction of cases as the angle climbs.

Above a hundred and fifty they nearly cancel. Six hundred and fifteen pairs are here, and on these an audit reporting two departures of comparable size is reporting two things that between them do almost nothing.

Which regime a given pair is in cannot be guessed from the two magnitudes, and it cannot be guessed from the physiology either: the lens and the macular pigment are both short-wavelength filters, which suggests they should always reinforce, and they reach a hundred and sixty degrees on some surfaces because their absorption bands have different shapes and the metric weighs the difference between the shapes.

The only way to know is to compute it, and computing it is the same two vectors and one matrix on every surface.

The rule as a formula a specification could carry #

The composition rule is short enough to state in a document.

Two departures of stated size, at a stated angle, compose to the root of the sum of their squares plus twice their product times the cosine of the angle. That is the cosine rule, it is older than colorimetry, and the only thing new here is which angle it wants: not the angle between the two deviations in tristimulus values, which is available and is the wrong one, but the angle in the local metric at the colour where the two land.

The two angles are not close. Over the pairs measured here the Euclidean angle and the metric angle differ by a median of twenty-four degrees and by as much as a hundred and thirty, which is what an axis ratio of up to thirty does to angles. Using the Euclidean angle is better than using no angle and it is not the rule: it leaves a median error of 13.3 per cent against the metric angle’s 1.2.

So a specification that wanted this would need three numbers per pair rather than two, and the third is the one an audit is most likely to think it already has. It does not: nothing in a table of sizes carries a direction, and nothing in a table of tristimulus deviations carries the metric they will be read through.

Which pairs reinforce, and which undo each other #

Aggregating by pair rather than by surface gives a table worth reading on its own, because the medians are stable and they have physiology in them.

The most reinforcing pair is the lens against the pigment peaks, at a median of 37 degrees, cancelling on two surfaces of forty-two. A yellowed lens removes short-wavelength light; a long-wavelength cone peak shifted upwards removes short-wavelength sensitivity from the channel that has least of it. The two act on the same end of the spectrum and mostly add.

The most opposing pair is the field size against the macular pigment, at a median of 162 degrees, cancelling on forty-one surfaces of forty-two and never falling below 112. That one is not a coincidence and it is not a discovery either — it is the audit’s own construction returning: a ten-degree field is entered as two changes, and one of the two is less macular pigment. So the field-size departure contains a macular departure with the sign reversed, and putting the two together removes most of both.

That is a defect in the audit’s parameterisation rather than in anybody’s eye, and the angle is what makes it visible. A table of six magnitudes cannot say that two of its rows overlap; a table of fifteen angles says it in one entry, and says how much.

The middle of the table is where the interesting cases are. The macular pigment against the pigment peaks sits at 60 degrees and cancels on eleven surfaces of forty-two — a pair that mostly reinforces and sometimes does not, with nothing in the two numbers to say which surface is which. The rods against the pigment peaks sit at 119 and cancel on twenty-six, which is a pair that mostly undoes itself.

Six of the fifteen pairs have a median angle above 120 degrees. An audit whose terms were independent in the sense quadrature requires would have them scattered around ninety with no structure, and they are not scattered around anything.

What this changes about a worst case #

The place where composition matters most is a bound, and a bound built from magnitudes is not a bound.

This collection has twice found the same error in its own worst cases: a maximum over a sample of a set is not the maximum of the set, and a listed worst case is not the worst case. The composition problem is the third member of that family and the largest of the three.

A worst case over six departures built in quadrature comes to a number. Built with the angles it comes to a different number, larger on the pairs that align and smaller on the pairs that oppose, and which of those a particular observer is in is decided by the surface. There is no scalar worst case over six departures, because the six do not have a fixed geometry — the same objection a mean raises against a worst case; there is a worst case per colour, and the spread of those is what a specification would need.

The deviations compose by ordinary vector addition and always have. Everything difficult here enters at the last step, when three numbers become one, and the difficulty is entirely a property of that step.

What was computed, and how #

The pairs are every unordered pair of the six observer departures, evaluated on every surface in the family: fifteen pairs on a hundred and sixty-eight surfaces, two thousand five hundred and twenty in all, with six hundred and thirty of them on the audit’s own forty-two.

Each deviation is the difference between two observers’ adapted tristimulus readings of the same sample under the same light, formed exactly as the audit forms it. The composition is done on the deviations and priced once — so nothing here is a claim about how two changes to an eye interact physiologically, which is a question about pigments and is not this question. Two pigment changes applied together do not give the sum of the two deviations, because absorptances multiply. That is a separate and larger subject and it is not what this measures.

The metric is estimated by finite differences at a step of two parts in a thousand of the sample’s own luminance. Halving that step changes the reported angles by under a tenth of a degree and changes no conclusion; making it much smaller starts to measure floating-point noise in the difference formula’s own square roots.

Where the model stops #

The local form is a linearisation and the departures are finite. The 1.2 per cent residual is that, and it grows with the size of the deviation: at ten colour differences rather than one the cosine rule is out by nearer a tenth. Nothing here proposes it as a general composition law, only as the correct next term after the one everybody uses.

ΔE₀₀ is not smooth everywhere — its hue rotation term has places where the formula is not smooth — and near those the local form is not a good description of anything. The pairs measured here avoid them by accident rather than by design, and a pair straddling a discontinuity would be reported wrongly by both rules and by this essay.

And the angle is a property of the colour, so it inherits the whole of the previous essay’s problem: the audit’s surfaces are pale, the angles are measured on them, and what the distribution looks like over a set of real reflectances is not known.

The generalisation #

The habit is about what a table of magnitudes cannot be asked.

A sensitivity analysis, an error budget, an ablation study — anything that varies one input at a time and records how much the output moved — produces a column of sizes. The column is genuinely useful and it supports exactly one operation, which is ranking. It does not support addition, and the arithmetic everybody performs on it is addition in quadrature, which is a statement about geometry that the column does not contain.

The move is to keep the direction. It is usually free: the thing that was varied produced a vector before somebody took its length, and storing the vector rather than the length costs the width of the output and buys every composition anybody will later want.

The failure mode is that the composed estimate is not merely imprecise but wrong in a direction that flatters. Quadrature over six terms is conservative when they align and optimistic when they oppose, and the optimistic half is invisible: nothing in the table says which pairs cancel, so nothing warns the reader that the bound they have built is not one.

Who found it, and when #

The cosine rule in a Riemannian metric is nineteenth-century mathematics, and the observation that colour difference formulae define such a metric locally is standard: the whole line-element tradition in colorimetry, from Helmholtz through Schrödinger to Stiles, is built on it, and MacAdam’s ellipses are that metric measured.

What is not standard is applying it to the composition of audit terms rather than of stimuli. Error budgets in colour management add their stages in quadrature and this collection’s own three-stage budget does so. No source consulted here computes the angles, and the reason appears to be that budgets are assembled from numbers rather than from the calculations that produced them, by which point the directions are gone.

Where the ladder goes next #

The price and the angle both come from one function, and that function has a feature nobody has had to look at: CIELAB’s compression is not a cube root. Below a stated luminance it is a straight line, spliced on so the whole thing has a finite slope at zero — and every black anybody delivers is inside the straight piece, where the compression is not compressing at all.