A confusion point is a missing pigment

Colour matching cannot say what the cones are. What can is a population that is missing one of them.

The claim #

Three confusion points supply six numbers, three choices of unit supply three more, and nine is exactly the dimension of the freedom that matching data leave open. The construction is a determination rather than a fit: there is nothing to minimise, no residual, and no parameter left over.

• A dichromat’s confusions are collinear, and the lines converge on a single point per class — a fact about the algebra rather than an empirical regularity, once trichromatic matching is granted.
• The point is the direction the missing receptor would have measured, which is the cross product of the two surviving rows of the matrix carrying tristimulus values to cone responses.
• Two of the three points are outside the diagram. The deuteranope’s sits at (1.40, −0.40), which is not a colour and does not need to be.
• Running it backwards is a test any matrix can be given. Smith–Pokorny returns the three published points to 1.9 × 10⁻⁴; Hunt–Pointer–Estévez misses the deuteranope’s by 1.28, CAT16 by 1.45 and CAT02 by 4.09.
• And the determination is exact, checked as the rank of a 9 × 9 Jacobian rather than as arithmetic.

Why the lines converge #

Let M be the matrix carrying tristimulus values to cone responses, with rows mL, mM, mS. A protanope has no long-wave cone, so two stimuli are indistinguishable when the other two rows report the same thing about both — that is, when their difference is annihilated by mM and by mS.

Two linear conditions on a three-dimensional space leave a line. So the set of differences a protanope is blind to is one direction, and it is the cross product mM × mS. Any two stimuli differing by a multiple of that direction are confused, whatever they are.

In a chromaticity diagram, adding a multiple of a fixed direction to a stimulus traces a straight line, and every such line passes through the chromaticity of the direction itself. That is the convergence: not twenty-five separate observations that happen to line up, but one direction seen from twenty-five starting points.

What the collapse actually is #

It is worth being exact about the geometry, because the word confusion suggests an error and there is no error anywhere in it.

A trichromat’s colour space is three-dimensional: three receptor classes, three numbers, and a stimulus is a point. A dichromat’s is two-dimensional in the same sense — two classes, two numbers — and the map from the first to the second is a linear projection whose kernel is exactly one direction. Every fibre of that projection is a line, and a line’s worth of stimuli arrives at the dichromat as one colour. Nothing is being mistaken; a measurement is being made with one fewer instrument.

The chromaticity diagram is a projective picture of the three-dimensional space — rays through the origin drawn as points — and a family of parallel lines in the three-dimensional space becomes a family of lines through one point in the picture. That point is where the kernel direction itself lands. So the convergence is a property of the drawing as much as of the eye: the same collapse plotted in a space where luminance is kept would be a set of parallel planes and would look like nothing at all.

Running the construction backwards #

The relation between rows and directions is symmetric, which is what makes the construction available.

If dP = mM × mS, then mM · dP = 0 and mS · dP = 0. Applying the same reasoning to all three:

dP ∝ mM × mS and dD ∝ mL × mS and dT ∝ mL × mM

so each row is orthogonal to two of the three directions:

mL ∝ dD × dT and mM ∝ dP × dT and mS ∝ dP × dD

Each row comes out determined up to a scale. Three rows, three scales, and the six numbers of the three points: nine.

The scales are the part that is a convention rather than a measurement, and any consistent choice will do. The one used here is that the adopting white maps to one in all three channels, which is the normalisation Hunt–Pointer–Estévez uses, so that the two matrices differ in their directions and in nothing else. Other conventions in the literature normalise so that the long- and medium-wave responses sum to the luminous efficiency function, or so that each curve peaks at one. None of them is a claim about an eye, and every published set of fundamentals carries one.

The test any matrix can be given #

Because the construction runs both ways, a matrix does not have to announce its confusion points. It has them, and they can be read off.

Two things are worth noticing about the table before the individual rows. The first is that a matrix can miss badly on one point and closely on another: Hunt–Pointer–Estévez is within 0.007 of the tritan point and 1.28 from the deutan one, because its short-wave row depends on Z alone — a strong and nearly correct structural assumption — while its long- and medium-wave rows were chosen for their behaviour under adaptation. The second is that the misses are not small in any of the columns where they are non-zero. A chromaticity difference of 0.13 is roughly the distance from a warm white to a cool one, and 1.28 is off the diagram altogether.

The row that matters most is the third and fourth. CAT02 and CAT16 are the axes appearance models adapt in, and they are routinely described as cone responses — the letters at the ends of their rows are L, M and S. On this test CAT16 puts the deuteranope’s confusion point 1.45 away from the measured one and CAT02 puts it 4.09 away. Those axes were fitted to corresponding-colour experiments, which is a wholly respectable thing to fit to and is not this measurement, and what that costs an appearance model is a question of its own.

The Bradford transform, still embedded in every ICC workflow, does worse again and has entries that no receptor sensitivity could have, since a receptor’s response to a light cannot be negative where the light is present.

Three of the six numbers are structure, not measurement #

The count is stated as six numbers from three points and three from a convention. Reading the three points as they are printed says something sharper: three of those six numbers are not measurements at all.

Three exact zeros in six quoted numbers, at four decimal places, is not a run of luck. Each is a structural assumption showing through, and each has a name.

The two points on x + y = 1 say the short-wave cone depends on Z alone. A chromaticity with x + y = 1 has z = 0, so both the protan and the deutan confusion directions have no Z component. The protan direction is mM × mS and the deutan is mL × mS, and both are forced to have zero Z exactly when mS is proportional to (0, 0, 1). One assumption, two of the six numbers.

The tritan point on y = 0 says the short-wave cone contributes nothing to luminance. The tritan direction is mL × mM, and it is annihilated by both of those rows; its Y component vanishes exactly when ȳ lies in the span of mL and mM — which is the statement that luminance is a combination of the long- and medium-wave cones with no S term. One more assumption, one more of the six.

So the honest count is three measured numbers, three structural impositions, and three unit conventions. Nine is still nine and the determination is still exact; what changes is which parts of it a revised dichromat experiment could move. Only 0.7465, 1.4000 and 0.1748 are on the table.

Which makes the backwards test less discriminating than it looks #

That has a direct consequence for the table of five matrices, and it explains a row the essay finds puzzling.

Any matrix built on the same two assumptions inherits the same three zeros, so it passes three of the six comparisons before anything is compared. Hunt–Pointer–Estévez is the worked example. Its short-wave row is exactly (0, 0, 1), so its implied protan and deutan points land on x + y = 1 to the last digit; its long- and medium-wave rows span ȳ, so its implied tritan point comes out at y = −7 × 10⁻⁶.

Computing all three from the published coefficients:

which reproduces the three figures quoted above to the digits they are quoted at. And every miss is exactly one-dimensional. The protan error is (+0.0909, −0.0909) and the deutan error is (+0.9022, −0.9022): both lie precisely along the direction (1, −1), which is the line x + y = 1 that the S-row assumption pins the point to. Neither has any component across it, and neither could.

So the essay’s explanation of the HPE row — that it is close on tritan because its short-wave row depends on Z alone — has the right ingredient attached to the wrong point. The Z-only S row is what puts the protan and deutan points on their line; what puts the tritan point at y = 0 is the separate normalisation making luminance a combination of L and M. Both are real, both are load-bearing, and they act on different rows of the table.

The test remains worth running and its resolution is lower than the numbers suggest. Of the six degrees of freedom in three confusion points, a matrix built on the standard structural assumptions has three of them fixed for free and is being examined on the other three. What the table measures is therefore how well a matrix’s directions within those constraints match the dichromat evidence — a three-number comparison presented as a six-number one, and a good deal more demanding on the two rows that were fitted for something else than on the one that was not fitted at all.

What was computed, and how #

The three points used here are the values the Smith–Pokorny fundamentals are built on: (0.7465, 0.2535), (1.4000, −0.4000) and (0.1748, 0.0000). They are quoted, because they are a measurement of people and this collection’s fourth invariant is that measurements are quoted and everything downstream of them is computed.

The check that the algebra is the field’s algebra is the round trip. Running impliedCopunctal on the published Smith–Pokorny matrix returns those three points to within 1.9 × 10⁻⁴, which is the rounding in the published coefficients. That is not a test of the points and it is not a test of the fundamentals; it is a test that the cross-product construction written here is the one the literature used, and it is the only such test available.

The construction refuses the degenerate case rather than returning something plausible. Three confusion points lying on one line determine no matrix at all — the three cross products become parallel and the determinant vanishes — and koenigMatrix throws on it. The refusal is asserted, on this site’s standing rule that a check which has never rejected anything proves nothing.

Where the model stops #

The confusion points are a smaller measurement than the matching functions. The 1931 functions come from seventeen observers; the copunctal points come from studies of dichromats numbering in the tens, with the tritan point resting on the fewest people. So the nine numbers are pinned by the least well-supported part of the evidence, and a revision to the tritan point would move the short-wave row of every set of fundamentals derived this way.

Dichromacy is being treated as trichromacy minus one receptor. That is the standard reduction hypothesis and it is not free: it assumes the two surviving classes in a dichromat have the same sensitivities they have in a trichromat. There is good evidence for it and it is an assumption, and it is the assumption the whole construction rests on. A dichromat whose surviving pigments were shifted would supply a confusion point about their eye rather than about the standard observer’s.

The reduction is also what makes the simulation drawable at all. A projection needs a plane to project onto, and the plane is chosen so that the stimuli a dichromat and a trichromat are known to agree about — the neutral axis and two anchor wavelengths — are fixed by it. Those anchors are a further empirical input, quoted from the unilateral dichromat literature, and they are the reason this site’s simulations name their method and their severity rather than claiming to show what somebody sees.

And the anomalous trichromats are not in this picture at all. The commonest colour vision deficiencies are anomalous rather than absent — a shifted pigment rather than a missing one — and they have no confusion point, because nothing is being annihilated. What this site simulates at a severity below one is an interpolation toward the dichromat limit and is explicitly not a model of anomalous trichromacy.

Who found it, and when #

Arthur König and Conrad Dieterici published the derivation in the 1880s and 1890s, and the reasoning is unchanged. What has changed is the data: König’s dichromat measurements were superseded, Judd and then Vos corrected the underlying matching functions in the blue, and Smith and Pokorny in 1975 and Stockman and Sharpe in 2000 produced the sets in use now.

What has also changed is the vocabulary. König’s word was Grundempfindungen, fundamental sensations, and the modern cone fundamentals is a direct descendant of it — a name from an era in which the receptors had not been measured and their sensitivities were an inference from behaviour. That is exactly what they still are in this construction. The direct measurements arrived much later, from microspectrophotometry of individual receptors and then from genetics, and they agree with the inferred curves well enough to make the inference respectable and not so well as to make it redundant.

The persistent oddity is how rarely the structure is stated when the result is quoted. A textbook prints the cone fundamentals as three curves beside the matching functions, with a matrix between them, and the matrix arrives as a number rather than as a derivation. The three points are where it comes from, and they are a measurement of a different population than the one the matching functions came from — which is a much more interesting fact about the standard observer than the curves themselves.

The generalisation #

The shape is a parameter that one experiment cannot reach and a second, apparently unrelated, determines exactly.

It is worth being precise about why the second experiment works, because the reason is not that dichromats are more informative. It is that removing a receptor turns a three-dimensional matching space into a two-dimensional one, and the direction that was lost is the thing the trichromatic experiment integrates over. A measurement that collapses a dimension reports on the dimension it collapsed.

That is a usable tactic. Where a model has an unidentifiable direction, the experiment that fixes it is often one that breaks the symmetry responsible — not a more precise version of the original measurement, which will produce a better-determined point on the same orbit and nothing else.

The same shape appears twice more in this collection and both times the second experiment is the one that looks least like the first. A highlight recovers the lamp because a specular reflection is the one part of a scene that is not multiplied by a reflectance, so it breaks the symmetry between a bright surface and a bright light. An instrument’s own geometry separates the interface reflection from the body reflection because it changes what is collected rather than what is illuminated. In each case the informative measurement is the one that removes a degeneracy rather than the one that reduces a variance, and the two are usually confused because both are described as better data.

What the construction is not #

Two readings of this essay would be over-claims and both are easy to fall into.

It is not a derivation of the cone fundamentals from first principles. Nothing here explains why the receptors have the sensitivities they have, and the construction takes two experiments as given and combines them. What it supplies is a count — that the two experiments together determine the answer exactly, with nothing over — and a route, which is the geometry above.

And it is not a claim that the standard fundamentals are the only defensible ones. Different laboratories impose different normalisations, use slightly different confusion points, and start from Judd-corrected or uncorrected matching functions, and the results differ by amounts that matter for some purposes and not for others. The construction says those are choices about units and about which measurements to trust; it does not adjudicate between them.

What the construction does do, and what nothing else here does, is make the choices visible and countable. Nine numbers, six from a dichromat experiment and three from a convention, is a much more useful description of a set of published curves than three plotted lines.

Where the ladder goes next #

The freedom the confusion points close is the freedom to change coordinates, and on a chromaticity diagram that is a projective map. What such a map preserves and what it does not decides which of this collection’s own claims are about the eye and which are about the paper — and the most quoted of them turns out to be about the paper.