The eye is a picture surface too

Which rule a fisheye obeys, from straightness alone put a plumb-line fit to a photograph and asked which of four candidate rules — equidistant, equal-area, stereographic, orthographic — describes the lens that took it. Nothing in that fit refers to glass, or to a camera at all. It is given marks, told which sets came from edges that were straight, and asked which rule straightens them best.

There is another optical instrument sitting a few centimetres from the reader of this sentence, and the same fit can be put to it. The eye is a place, not a point already established that an eye has an entrance pupil rather than a mathematical point, which is one way this site has treated vision as an instrument subject to its own measurements rather than as a metaphor. This essay goes further: a nearly spherical retina sitting behind an off-centre nodal point is, geometrically, a picture surface exactly as a camera’s sensor or a planetarium dome is one, and every measurement this collection has built for a picture surface can be run on it unmodified.

Running that battery gives four answers, in order: the eye’s own rule, how far a straight line’s true image departs from a straight line drawn on the retina, how unevenly the eye spends its own resolution, and — the result this site is built to make — that a flat picture at its own correct viewing distance leaves exactly the same trace on that retina as the world it depicts. None of this is a claim about what a straight edge looks like to whoever owns the eye. That is a question about perception, this collection does not compute perception, and the geometric result stands on its own regardless of how the answer feels.

Treating an eye this way is not a metaphor borrowed for colour. A retina behind a nodal point satisfies every assumption the site’s picture-surface measurements already require: a single centre every ray passes through, a stated surface the rays are cut by, and a well-defined map from a world direction to a point of that surface. Nothing in the rule-fitting method or the six-surface battery elsewhere in this collection refers to glass, a sensor or a focal-plane shutter. Both refer only to a centre and a surface, and an eyeball supplies both without needing to be told it is being asked to.

A circle, and not a great one #

Every ray from a straight world line that reaches this eye passes through one point — the nodal point — so the rays all lie in a single plane, and the mark that plane leaves on a sphere is a circle, exactly, to 2.6 × 10⁻¹⁵ millimetres of planarity in the schematic eye drawn above. That is the same fact an angle is a cross-ratio and the projective essays elsewhere on this site keep finding in different guises: a bundle of rays through a common point is flat, and cutting a flat bundle with any surface gives a conic.

What makes the circle above not a great circle is worth stating precisely, because it is the one fact a naive drawing of “an eye as a sphere” gets wrong. A great circle’s plane has to pass through the sphere’s own centre, and the nodal point in a schematic eye does not sit at that centre — it sits 4.7 millimetres in front of it, toward the cornea. A world line whose plane already contains the optical axis still traces a great circle, because that plane holds every point on the axis including the centre. Every other line does not, and the hero figure’s own line, run through a nodal point offset that far, returns a circle of angular radius 70.2° rather than ninety — the missing 19.8° is the whole effect of the offset, on this one line, and it would be exactly zero if the nodal point sat at the sphere’s centre instead of in front of it.

The nodal point is doing exactly the job the eye is a place, not a point already assigned an entrance pupil: it is the single location a bundle of rays is required to pass through before anything downstream can be treated as a projection at all. That essay measured what changes when a lens’s entrance pupil is not where a simple model assumes it to be. Here the same displacement — a single point standing in for glass, offset from the sphere it sits in front of — is not a defect to be corrected for; it is the anatomical fact that turns every retinal image of a line into a small circle rather than a great one, and the 19.8° gap above is what that fact costs on one particular line. A circle’s plane not passing through a sphere’s centre is an ordinary statement in solid geometry, the same one behind the circle whose centre moves for a plane section of a cone; what is specific to an eye is only which point the bundle is forced through and how far that point sits from where a naive drawing would put it.

Which rule the four-way fit assigns to a retina #

The fit from the previous essay does not know or care whether the marks it is given came from a lens or an eyeball. Point it at the schematic eye’s own geometry and ask the same question.

The eye is usually drawn as the equidistant fisheye — the textbook diagram puts a straight radial scale on the visual field, because that is the simplest thing to draw and the easiest to reason about — and that habit is worth naming before it is corrected, since the correction is the more interesting fact.

The eye is close to the equal-area rule and nowhere near the equidistant one, which is worth sitting with for a moment because the two claims sound similar and are not. Equidistant means a straight radial scale — double the eccentricity, double the distance from the middle of the field. Equal-area means a straight area scale — a patch of sky subtending a fixed solid angle prints a fixed patch of retina wherever it falls — which is the property a mirror ball also turns out to have, by a wholly different route. Two very different physical instruments, a curved mirror and a curved retina, land on the same rule, and neither was built by anyone consulting the other.

That agreement is worth one more caution before it is used for anything: moving the nodal distance from the anatomical 16.7 millimetres to 18 already widens the equal-area residual by more than a factor of ten. The eye is close to equal-area for this eye’s own proportions, and the fit above is what says exactly how sensitive that closeness is — a coincidence of scale rather than a law the eye is built to obey, and stated as one because the fit that would have contradicted it was actually run.

That sensitivity is itself informative about what kind of claim “the eye is equal-area” can support. A designed instrument that obeys a rule by construction — a lens ground to a stated formula, or the swing-lens camera further down this essay — keeps obeying it under a small manufacturing tolerance, because the rule was the target the tolerance was drawn around. An anatomical proportion that happens to land near a rule has no such tolerance band; it is simply wherever evolution and growth left it, and the fit’s own sensitivity to a millimetre is the honest report of how much weight that particular coincidence can bear. Counting cloud by counting pixels leans on the equal-area rule for a different purpose entirely — turning a picture’s own pixel counts into a solid-angle measurement — and it is worth noting that the machinery there is exact by definition for a sensor built to the rule, in exactly the way the fit above is inexact for a retina that merely resembles one.

What the geometry does, and what it does not claim about looking #

The retina is curved and a straight world line’s image on it is a curve rather than a straight line. What that costs, measured rather than asserted, is the next question — and it is also where the loudest limit in this essay belongs, stated before the number rather than after it: whether a straight edge looks bent to whoever owns this eye is a question about perception, and this collection does not compute perception. It computes where light lands. What follows is only that.

Nineteen degrees of visual angle is a large number as angles on a retina go, and the honest reading of it is narrow. It says a straight world line’s image departs from a straight line drawn on the flattened retinal picture by that much, as a matter of where photons land. It says nothing about the downstream processing that turns a pattern of retinal illumination into an experience of a straight or a curved edge, which is a question for a different field entirely and one this site has no machinery to touch. The two questions get confused easily because they share a vocabulary — “bent,” “curved,” “distorted” — and the geometry earns the right to use that vocabulary about light while owing none of it to the separate claim about seeing.

No surface keeps everything already established, for cameras and screens rather than eyes, that straightness is a property exactly one of the six named picture surfaces has, and every curved one gives it up entirely rather than merely weakening it. The retina is a curved surface in that same census, so the nineteen-degree sag is not a surprising defect to be explained away; it is the unavoidable price stereographic keeps every angle already showed every curved surface pays somewhere, applied to a sphere that happens to be made of rod and cone cells rather than glass or a screen. What is specific to the eye is only that the price is paid on an instrument nobody chose, rather than one a designer selected among six known options.

How unevenly the eye spends what it has #

Where a surface spends its pixels measured the same question for the six named picture surfaces this collection uses for cameras and screens: how much resolution a surface gives a patch of world depends on where that patch sits, and the flat plane is the worst offender by a wide margin. Running the identical battery on the retina places it precisely.

A square degree spending very nearly the same area everywhere is the retina behaving like a good instrument by the resolution measure, spreading its finite supply of receptors evenly across the field it covers rather than lavishing them at the centre and starving the edge, or the reverse. The anisotropy of 1.911 is the separate, unavoidable cost every curved surface in this collection pays for being curved: nothing that keeps a square degree’s area fixed everywhere can also keep every angle fixed everywhere, which is the same trade conformal is not undistorted already priced for the little-planet projection. The eye buys even coverage and spends the difference on shape.

An instrument that is exactly a surface, and one that is only nearly #

The equal-area finding above is a fit — a best available match, with a residual and a stated sensitivity to the anatomical numbers going into it. It is worth setting beside a case elsewhere in this collection where an instrument is not nearly a named surface but exactly one, so the difference between “coincidence of proportions” and “identity by construction” is visible rather than asserted.

The swing-lens camera is the cylinder by construction: the film is bent into a circle about the pupil as a matter of engineering, so the agreement with the named surface is not a fit at all — there is no free parameter and nothing to be sensitive to. The eye’s agreement with the equal-area rule is the opposite kind of fact: no evolutionary pressure bent a retina into a sphere in order to be equal-area, the fit needed one free focal length to align the two, and the previous section already showed how quickly that agreement degrades when the anatomical numbers move by a millimetre. Both instruments are picture surfaces. Only one of them is a named surface for a reason a designer intended.

The claim this site is built to make, put on a sphere #

Every essay in this collection returns eventually to one arithmetic fact: a picture is correct from a specific, computable distance, and the geometry does not care whether “correct” sounds like an unusually strong word for a photograph. Restating that claim on a curved retina rather than on the usual flat comparison is the point of building this battery at all.

That is the viewing-distance claim with nowhere left to hide: not “the picture looks about right from here,” but that the pattern of light reaching a retinal sphere from the correct distance is, to fifteen digits, the identical pattern the depicted world would have sent it. A photograph and the scene it depicts are not usually alike in any strong sense — they differ in size, in medium, in nearly everything a viewer could point to — and at one specific distance from one specific picture they become, as far as this measurement can tell, the same input to an eye.

The screen sets the distance made the equivalent case for a flat display rather than a curved retina, and the two results are the same fact carried across a different second surface: a picture is a projection onto a sensor, and looking at it is a second projection into a room, and the second projection has a correct standing point exactly as the first has a correct camera position. What this essay adds is that the second surface need not be flat for the argument to close. The retina is curved, the world’s light already arrives on it curved by the same nodal-point geometry the earlier sections measured, and the flat sheet’s light — routed through an ordinary pinhole and then through the same curved retina — arrives bent in exactly the matching way, at exactly one distance. Curving the receiving surface does not weaken the claim; it is carried through it unchanged, because both the sheet’s light and the world’s light pass through the same final step.

That second figure is the control the first one needs to mean anything. A method that reported “the same arc” regardless of where the eye stood would be reporting an identity rather than a measurement, the way a stopped clock reads the right time twice a day for no reason connected to time. Move the eye by less than half again its correct distance and the agreement does not merely degrade — it opens by nearly ten degrees of visual angle, thousands of micrometres of retina, from a starting point of exact coincidence. The viewing-distance claim is not a rounding statement about pictures being roughly convincing near enough to the right spot; it is a knife-edge, and this is what falls off the wrong side of it.

What the schematic model leaves out #

The eye used throughout this essay is the reduced eye of physiological optics: one spherical refracting surface, one nodal point, one retinal sphere, chosen because it is the simplest object for which “nodal point” and “retinal sphere” are exact rather than approximate terms. A real eye has a cornea and a lens working together, an aspheric rather than perfectly spherical retina, chromatic and off-axis aberrations that grow toward the periphery, and a fovea that is a small pit rather than a point — none of which this essay’s arithmetic represents, and all of which would move the numbers above without changing which side of the viewing-distance knife-edge a reader stands on.

What a ray does at a surface computes the one further step this essay skips entirely: a real eye refracts before an image ever reaches a retina, bending each ray at the cornea and the lens rather than passing it through a single nodal point for free. The reduced-eye model used here folds all of that refraction into one equivalent point precisely so the rest of the site’s picture-surface machinery — built for a pinhole, not a compound lens — can be run on it without modification. That is a genuine simplification and not a free one: it is exact for paraxial rays near the optical axis and increasingly approximate as the eccentricity grows, which is also, not coincidentally, the region where the numbers in this essay are largest. A model built to make the eye tractable is not the same claim as a model built to make it precise, and the reduced eye is unambiguously the first.

Nor does anything here touch how a retinal image becomes a percept. The retina is not read out like a camera sensor in one pass; it is sampled unevenly, compressed, and reconstructed by machinery this collection has no access to and no business modelling. The geometric result — a straight line’s retinal trace is exactly a circle, and a correctly viewed picture reproduces it exactly — is true regardless of any of that downstream processing, and it would remain true even if perception turned out to correct for the retina’s curvature so completely that nothing about it were ever noticed. Geometry and perception are answers to different questions, and this essay only had the machinery for one of them.

The same fit, run on a picture nobody photographed #

The four-way rule fit and the retinal battery both treat their subject as marks and a surface, with no interest in whether a lens, an eyeball or something else produced them. The arcs the five-point construction actually draws puts the identical fit to a third kind of object again — a draughtsman’s curvilinear-perspective construction, which was never photographed by anything and never passed through an eye, and asks the same question of it: which rule, if any, does this drawing actually obey. That the method transfers cleanly across a lens, a retina and a ruled construction is the real content of “from straightness alone” that the earlier essay’s title claims — the fit does not know what it is being shown, only whether one candidate rule straightens the marks and the others do not, and an eyeball turns out to be exactly the kind of object that question can be put to.