A point about the pigments that remain

The protanope’s confusion point is the place a protanope’s confusion lines meet, and it is not a fact about the protanope’s missing pigment. Moving that pigment does not move the point at all, and the “at all” is exact.

The claim #

Each dichromat’s confusion point is determined by the two receptors that remain, and is exactly independent of the one that is absent.

• The protanope’s point is unmoved by the L pigment, to 1 × 10⁻¹⁶ in chromaticity, over a sweep of ±30 nanometres.
• The deuteranope’s is unmoved by the M pigment, and the tritanope’s by the S pigment, to the same precision.
• Each moves a great deal under the two it depends on — the protanope’s point moves 0.13 in chromaticity for an eight-nanometre shift of the M pigment.
• The reason is linear algebra rather than physiology: a confusion point is the null direction of two rows of a matrix, and scaling the third row does not move it.
• And the point runs to infinity where the two long-wave pigments coincide, which is the correct answer rather than a numerical difficulty.

What a confusion point is #

A dichromat has two cone classes rather than three. Any two stimuli that produce the same pair of responses are indistinguishable to them, so the set of stimuli confused with a given one is a line in chromaticity — a confusion line — and all the lines of one class of dichromat meet at a single point, the copunctal point.

The point has a physical reading and it is the reading that makes it interesting: it is the chromaticity of a stimulus that would excite only the missing receptor. A protanope cannot distinguish anything along a confusion line because the direction the L cone would have responded along is the direction they have no receptor for, and the copunctal point is where that direction crosses the chromaticity plane.

Which is exactly why the result below is surprising. The point is named for the missing receptor, is interpreted as the missing receptor’s direction, and is determined entirely by the other two.

The measurement #

Take the receptor basis as a 3×3 matrix B carrying tristimulus values to cone responses. Its three rows are the L, M and S fundamentals expressed in that coordinate system. The direction that excites only L is the direction in which the M and S responses are both zero — the null space of the M and S rows, which is a one-dimensional subspace whose chromaticity is the protan copunctal point.

The L row does not appear in that construction. Multiply it by any non-zero constant, or replace it with a different function altogether, and the null space of the other two rows is unchanged.

Shifting a pigment’s absorption peak changes that pigment’s row. So:

Exactly zero, three times. Not a small number that rounds to zero — the returned chromaticities are bit-identical across the sweep, because the arithmetic that produces them never touches the swept row.

Why this is not obvious #

Three reasons it reads as wrong the first time, and each is worth separating.

The point’s physical description sounds like a statement about the missing cone. The chromaticity that excites only L is a phrase whose subject is L. But “excites only L” is a constraint on M and S — it says the other two are silent — and a constraint that mentions a thing is not a constraint that depends on it.

The pigment’s own template does move the point when the template is read directly. Reading a confusion point straight off this collection’s pigment template gives a point that moves under everything, because the template’s three curves are fitted together and changing one changes the fit. That construction was measured a round ago and found to place the protan point at (0.99, 0.20) against a measured (0.7465, 0.2535) — which is why it is not the construction used. The transfer construction moves a member’s basis by the difference from the reference and keeps the published points for the reference member, and under that construction the independence is exact.

And the missing cone is not missing from the model, which is what a simulation of colour vision deficiency actually does and does not do here. Nothing here simulates a protanope by deleting a row. The point is computed from a trichromat’s basis and reports where a protanope’s confusion lines would meet, which is why the L row is present and, it turns out, irrelevant.

What the independence is worth is a number in two units at once, and both of them are in the same figure.

Where the point diverges #

Sweep the M pigment’s peak upward and the protan point runs away: (0.75, 0.25) at the measured value, (0.86, 0.14) at +12 nm, (1.00, 0.00) at +18, (1.30, −0.30) at +24.

At exactly +25 nanometres — the separation between the two long-wave pigment peaks — the M pigment sits on top of the L pigment. The two rows become parallel, their null space is no longer one-dimensional, and the direction that excites only L has no chromaticity at all.

The machinery throws rather than returning a number. That is the right behaviour and is worth saying so: the divergence is the correct answer, not a numerical difficulty. An observer whose two long-wave pigments are identical has no long/medium distinction, cannot have a protan confusion point in any meaningful sense, and a construction that returned a large finite number there would be lying about a singularity.

Past the pole the point reappears on the other side, which is the ordinary behaviour of a projective quantity and is the reason the search that restates the claim in nanometres uses a bounded scan rather than a root-finder — a solver let loose on this function would walk through the pole and return the other branch as an answer.

What it is good for #

Not a curiosity. It is what makes the restatement of a claim in nanometres possible, and it decides which nanometres.

The claim in question is that every published chromatic-adaptation transform implies a protan confusion point that is nowhere near the receptors’ own. Stated in standard deviations of a modelled population it depends on five widths this site declared. Stated as how far a pigment would have to move it depends on a pigment peak, which is measured — and the independence result decides which pigment: not the long-wave one the claim is nominally about, but the medium-wave one.

That is a case where a structural fact changes what a restatement can even say. A restatement in nanometres of the L pigment would have been impossible, since the answer is that no shift of the L pigment produces any change at all.

What it does not say #

It is not that the L cone is irrelevant to protanopia. A protanope’s vision differs from a trichromat’s in every way that having two receptors rather than three implies, and which receptor is absent decides a great deal — the luminous efficiency function, the wavelength of the neutral point, the ability to detect a red signal against foliage. The result is about one quantity: the chromaticity at which the confusion lines meet.

And it is not a statement about people with anomalous trichromacy. An anomalous trichromat has three pigments, one of them shifted, and their discrimination depends on the separation between the two long-wave pigments in exactly the way the divergence above suggests: as the two peaks approach, discrimination in the red–green direction collapses. That is a real and much-studied phenomenon and is where the population’s own spread comes from; this essay is about where a dichromat’s confusion lines meet, which is a different quantity that happens to be computed from the same matrix.

Why it was worth asserting rather than noting #

Because an exact zero is the strongest kind of check available and this collection has a rule about it.

The assertion sweeps each point against each pigment and requires two things at once: that the point does not move under the pigment its owner lacks, to machine precision, and that it does move under one of the two they have, by a stated minimum. Both halves are needed. A construction that had quietly stopped depending on the pigments at all — a cached matrix, a lost parameter — would satisfy the first half perfectly and fail the second, which is exactly the failure a one-sided check would pass.

That is this site’s standing habit stated in a particular place: an assertion that has never rejected anything proves nothing, so the assertion is written to make the boring failure fail, which is the habit this whole collection runs on.

The three points behave differently, and this says why #

The independence result explains a pattern this collection has published for four rounds without an account of it.

Two of the three confusion points sit outside any plausible population and the third does not. The tritan point is the one inside, and the tritan point is also the one whose spread across a population is largest — the cloud radius is 0.073 against the protan point’s 0.025.

Both follow from which pigments each point depends on.

The tritan point depends on L and M, the two long-wave pigments, which are separated by only 25 nanometres and whose fundamentals overlap enormously. Two nearly parallel rows have an ill-conditioned null space: small changes in either move the intersection a great deal, so the point is loosely determined, its population spread is wide, and anything within that wide spread is “inside the population”.

The protan point depends on M and S, which are separated by a hundred nanometres and overlap far less. That null space is well conditioned, the point is tightly determined, the population spread is narrow — and a published transform that misses it misses by many times a narrow radius.

So the same structural fact explains the ordering of the cloud radii and the ordering of the sigma-distances, and it does it without reference to any population width at all. A point determined by two nearly parallel receptors is a point nobody can place precisely, including this collection.

The deutan point behaves the same way under the same test, which is what makes the exact zero a property of the construction rather than of one dichromacy.

The conditioning, measured — and the point it does not explain #

That account rests on how nearly parallel two rows of the cone matrix are, and the angles are computable rather than descriptive.

The conditioning story is right about the pair it was told for. The two long-wave rows meet at 35 degrees where the other two pairs sit within four degrees of perpendicular, and an intersection’s sensitivity to a perturbation goes as one over the sine of the angle between the things intersecting — 1.73 against 1.00. That accounts for a factor of 1.72 of the 2.85 observed between the tritan cloud and the protan one, which is about half of it in logarithm; the rest is that the population’s widths move the two long-wave pigments more than they move S.

And it says nothing whatever about the deuteranope’s point, which is far the widest of the three. Its cloud radius is 0.391 — fifteen times the protan’s and five times the tritan’s, with a worst member out at 2.95 — and it sits on the second best conditioned pair on the list.

A second geometric factor supplies the ordering. A confusion point is a projective quantity: the null direction is divided by the sum of its three components to reach a chromaticity, and for the unit direction that sum is 1.268 for the protan point, 1.186 for the tritan and 0.687 for the deutan. A small sum is a point far outside the diagram, and it amplifies any wobble in the direction as the square of its reciprocal.

Multiplying the two factors gives 0.62 for the protan point, 1.23 for the tritan and 2.12 for the deutan, which orders the three cloud radii correctly and under-predicts their spread considerably — 3.4 predicted against 15.2 observed between the protan point and the deutan.

So the honest account has two geometric terms and a residual. A confusion point is imprecise when the two receptors that fix it are nearly parallel, and imprecise again when the point they fix lies far outside the diagram — and the deuteranope’s is the second case where the tritanope’s is the first. What is left over, a factor of about four on the deutan point, belongs to how far the population’s own widths move the long-wave row, which is not a geometric quantity at all.

Keeping the two mechanisms apart matters, because the argument above uses only the first and applies it to the two points where it happens to be the operative one. On the third it would predict the narrowest cloud in the table and the measurement returns the widest.

Where the model stops #

The exactness is a property of the construction, which builds a member’s basis by moving the published one and reads the points off it. That construction was chosen a round ago for good reasons — a template cannot place a point, so the template is used for differences and the measurement for the location — and the independence follows from its algebra.

A different construction gives a different answer. Reading the points directly off a fitted template, which is the construction anybody would write first, gives points that move under all three pigments, because the fit couples them. Neither construction is a measurement of anybody’s confusion lines; both are models — and a template cannot place a point — and this essay reports an exact property of the one this site uses and states that the property is not universal.

What the divergence is good for #

The pole is not only a numerical hazard to be scanned around. It is a statement about a real population.

As the medium-wave pigment’s peak approaches the long-wave one, the protan confusion point runs to infinity — which says the L-primary direction becomes undefined, which says the observer has no long/medium distinction left. That is not hypothetical. Anomalous trichromacy is exactly this situation part-way along: a person with a shifted M pigment has two long-wave pigments closer together than 25 nanometres, and their red–green discrimination degrades as the separation shrinks.

The curve gives the shape of that degradation in this collection’s own terms. At a separation of 25 nm the point sits at (0.75, 0.25); at 13 nm it is at (0.86, 0.14); at 7 nm it is at (1.00, 0.00); at 1 nm it is at (1.39, −0.39) and accelerating. The point’s distance from the diagram grows faster than the separation shrinks, which is the geometric statement of the same thing clinical measurement reports as discrimination collapsing faster than the pigment shift would suggest.

That connection is worth flagging and not overstating. This is a construction rather than a model of anomalous trichromacy — it moves a peak and leaves everything else, where a real anomalous observer differs in optical density and in the shape of the shifted pigment too. What it does say is that the divergence has a physical reading, and a construction whose singularity corresponds to a real limit is a construction behaving well.

Who found it, and when #

The algebra is König’s, in the sense that the fundamental-primary construction is his: a dichromat’s confusion lines converge on the missing fundamental’s primary, and the fundamental primaries are the columns of the inverse of the cone matrix. The observation that a column of an inverse depends on all rows but the corresponding one is a fact about matrices that every linear-algebra course contains and that nobody states in these words.

Its appearance here was accidental. The point was swept against the L pigment as the obvious first check while building the nanometre restatement, the answer came back as a stationary dot, and the sweep was assumed broken. It was not.

What the exact zero rules out #

An exact zero is a strong claim and it is worth saying what it forecloses, since the temptation with a surprising result is to look for the loophole.

It is not a small effect below the arithmetic’s resolution. The returned chromaticities are bit-identical across a ±30 nanometre sweep, not merely close. A small effect would show as a drift in the last digits; there is none, because the arithmetic producing the point never reads the swept row.

It is not an artefact of the sweep’s step size. The independence holds at every value tried and holds by construction — no step size can find a dependence that is not in the computation.

And it is not an artefact of the reference member. Sweeping from a different starting observer, with a different macular density and a different lens age, gives the same exact independence, because the argument is about which rows enter a null space and not about their values.

What it is contingent on is the construction: the transfer that moves a member’s basis away from the published one. Read the points straight off a fitted template and everything moves under everything, because the fit couples the three curves. So the honest statement is that this collection’s model has an exact property, that the property follows from a fact about matrices rather than about eyes, and that a different model of the same object does not have it.

Where the ladder goes next #

The independence result names the axis. What remains is to use it: the claim that rested on a declared population width, restated as a displacement in nanometres against a separation the eye fixes.