Where a surface spends its pixels

No surface keeps everything settles what the six named surfaces preserve: straightness, angle, area, and the fact that no surface holds all three. Every entry in that comparison is about fidelity — how much a surface distorts what it draws.

None of them is about price. A picture surface is a budget before it is anything else: a finite number of marks, spread over a set of directions, and the only decision it makes is where to put them. Two surfaces can be equally faithful in the sense of that earlier comparison and spend their marks so differently that one of them has no resolution left where the other has plenty.

This essay asks the second question. The answer is a factor of twenty-five between the extremes, at a field of view a real camera reaches.

The comparison has to hold something fixed, and it is not the focal length #

Before any number is quoted, the comparison needs a footing, and the obvious footing is wrong in a way that is worth measuring rather than asserting.

The natural thing to say is “compare the six at the same focal length”, or equivalently at the same image size. Do that and the pictures are not even the same size.

There is a second and sharper reason the focal-length footing says nothing. At a fixed focal length, five of the six surfaces score identically on the optical axis — 3.046 × 10⁻⁴ marks per square degree per focal length squared — because every surface here agrees with the plane to first order at the centre. A comparison on that footing has nothing at all to say about the axis, and everything it says off-axis is the area ratio again.

So the footing used throughout is different: the same number of marks, over the same cone of directions, for every surface. The gate keeps the wrong implementation, runs it, and requires it to fail, which is the only way to know that the right one is doing something.

The same world, thirty-six times over #

With the footing fixed, the cleanest statement of the result is not a number at all. Divide the cone into cells of equal solid angle and draw them.

Every cell in every panel is the same amount of world. What differs is the area each surface gives it, and that area is the resolution: a cell drawn twice as large has twice as many marks in it and records twice as much detail about the same piece of sky.

Read that way the flat plane’s behaviour stops being a curiosity about wide lenses and becomes the plainest fact about it. A rectilinear picture is a surface that spends almost nothing at the centre and lavishes marks on its corners — which is exactly backwards from what a photographer wants, since the subject is usually in the middle.

The distributions, as curves #

The mesh shows the shape. The curve puts numbers on it.

The 45° detail is worth pausing on, because it is the kind of thing that turns a measurement into a statement about the instrument rather than the subject. The cylinder is uniform in azimuth by construction; sweep along its equator and it reports perfect evenness, which is true along that one line and false of the surface. Sweeping at 45° crosses both of its axes and catches the anisotropy that a horizontal sweep is blind to.

The equal-area fisheye’s flatness is the opposite case and deserves the opposite caution: it is flat because its construction asserts flatness, not because the measurement discovered it. The 8.3 × 10⁻⁶ per cent is a check on the arithmetic — the number would move if the map or the integrator were wrong — rather than a finding about the surface. Reading it as a discovery would be reading a definition back as evidence.

The ranking, and the surface that is not where its reputation puts it #

Reduced to one number per surface, the ratio of edge spending to centre spending, the order holds a surprise in the middle of it.

The equidistant fisheye is the one to read carefully. It is described everywhere as the linear-in-angle projection, and it is: radius on the picture is exactly proportional to angle off the axis. A reader could reasonably conclude that it therefore spends evenly. It does not, and the reason is that a surface has two directions to spend in.

Radially the equidistant fisheye is flat by construction. Tangentially it is not: a ring of directions at angle θ from the axis has circumference proportional to sin θ in the world and to θ on the picture, so the tangential scale runs as θ/sin θ, which is one at the centre and grows to 1.30 at 70°. Area is the product of the two scales, so the surface that is exactly linear in one direction is 30 per cent uneven in area.

That is a real and specific correction to a description that is otherwise true, and it is the sort of thing this collection exists to notice: the property everybody quotes is a statement about one of the two directions, and the property a reader wants is about both.

Why the plane runs away and the others do not #

The ranking is not a list of empirical facts. Each surface’s number falls out of its own radial law in one line, and having the laws side by side explains why one of the six is qualitatively different from the rest.

A surface maps an angle θ off the axis to a radius r on the picture. Its radial scale is dr/dθ and its tangential scale is r/sin θ, because a ring of directions at θ has circumference proportional to sin θ in the world and to r on the page. The area scale is the product, and the spending ratio is that product at the edge over its value on the axis.

For the flat plane, r = f·tan θ. The radial scale is f·sec²θ and the tangential is f·tan θ/sin θ = f·sec θ, so the area scale runs as sec³θ — which is 25.0 at 70° and diverges at 90°. The divergence is the plane’s defining property, not a defect in it: the surface has to reach infinitely far to hold a direction at ninety degrees, and infinite reach for finite world is exactly what unbounded spending means. Every other surface here is bounded because every other surface reaches a half-turn or more at a finite radius.

For the equidistant fisheye, r = f·θ. The radial scale is f exactly — that is the selling point — and the tangential is f·θ/sin θ, which is the whole of the ×1.300. For the equal-area fisheye, r = 2f·sin(θ/2), and the two scales are reciprocal by construction, which is what the 8.3 × 10⁻⁶ per cent is checking.

So the six numbers are one formula evaluated six times, and the plane’s separation from the rest is a statement about which of them are bounded rather than about how badly any of them behaves.

What it looks like on a room #

The meshes and curves are the honest instruments. A picture of a room is what makes the size of the effect legible.

The comparison across those two figures is the whole essay in two pictures, and it is worth being exact about what changed and what did not. The room is the same, the mesh is the same amount of world per cell, and the number of marks is the same. Only the surface differs.

What this is not, and it is not distortion #

There is a strong temptation to read the flat plane’s ×25.0 as a statement about how wrong a rectilinear picture looks at the edges, and it is a different claim entirely.

The plane is the most faithful surface by the straightness measure — it is the unique surface that keeps every straight line straight — and the least even by the spending measure, by a wide margin. Those two facts are not in tension and neither is a consequence of the other. A stretched face at the corner of a wide rectilinear photograph is a fidelity effect, and it is the correct projection of that face from that eye; the resolution poured into the corner is a separate matter, and it is why the corner of such a picture is sharper than its centre while looking worse. Conformal is not undistorted makes the neighbouring point about the little planet, where a surface that preserves every angle still produces a picture nobody would call faithful.

So the two comparisons answer different questions and a reader has to know which one they are asking. What will be bent? is the earlier essay. Where will the detail be? is this one.

The measurement against the fleet’s other reading of the same surfaces #

It is worth setting this comparison beside the two the collection already has, because the three together are the whole of what choosing a surface decides, and no two of them rank the surfaces the same way.

Stereographic keeps every angle measures conformality and finds exactly one surface with a zero. The third column is area measures area distortion and finds exactly one with a zero, a different one. This essay measures where the marks land, and its zero — even spending — belongs to the same surface as the area column, for the obvious reason that spending is an area scale seen from the picture’s side rather than the world’s.

That last equivalence is worth stating rather than leaving implicit, because it means this essay adds nothing to the area comparison mathematically and a great deal to it practically. The area column says a shape drawn at the edge is the wrong size; the spending column says the same fact is a statement about how much detail the edge can hold. Same number, two consequences, and a photographer cares about the second while a cartographer cares about the first.

The straightness column stands alone and is the one that disagrees, since the flat plane is the only surface that keeps a straight line straight and is last on both of the others. There is no surface that is even, conformal and straight — and this essay’s contribution to that trade is that “even” has a precise price which can be read off a picture of a room.

What a reader can do with it #

The measurement is only worth making if it decides something, and it decides three things that come up.

Where the subject is. A picture surface should spend where the subject is, and most subjects are near the middle. Every fisheye is a different rule sets out the four laws a lens may obey; this measurement is what choosing between them costs in detail rather than in shape. On that criterion the flat plane is the worst available choice at wide fields and the equal-area fisheye the best, with the ordinary equidistant fisheye between them and closer to the good end.

What a field of view costs. The plane’s unevenness is not a fixed penalty; it is ×3.77 at 100°, ×24.995 at 70° measured to the corner of the cone used here, and ×57.68 at 150°, growing without bound as the field approaches a half-turn. Every one of those is the same surface. The choice of surface and the choice of field are not independent decisions, and a wide picture on a plane is a different object from a moderate one.

What a screen wants. The same arithmetic runs the other way for a display rather than a camera: a projector or a curved screen also has a finite number of marks and also has to decide where to put them, and the surface a screen wants is the essay that asks which surface a viewer at a stated seat is best served by. The spending measured here is the supply side of that question, and the two have to be read together — a picture taken on a surface that spends evenly and shown on one that does not has thrown the evenness away.

What a resampling costs. A reprojection from one surface to another is exact wherever both cover the direction, so nothing geometric is lost in one. What is lost is here: marks that one surface put in the middle cannot be conjured at the edge by another, and the resolution a picture has is fixed by the surface it was taken on.

Where the marks go on the surfaces that are not lenses #

Two of the six are not lens projections at all, and their spending has a different practical meaning worth separating out.

The cylinder is the panorama surface, and its ×2.396 is anisotropic in a way the single number hides. It is uniform in azimuth exactly — that is what makes it the right surface for a rotating rig — and it stretches only in elevation, as the secant of the angle above the horizon. So a cylindrical panorama has even resolution all the way round and loses it upward, which is why what a 360 photograph is can hold the whole horizon at one quality and cannot hold the zenith at all.

The equirectangular surface is the storage format rather than a picture anybody looks at, and its ×1.338 over this cone badly understates it, because the cone stops at 70° and the surface’s trouble is at the pole. A pole is a line measures what happens there: one direction is sent to a whole edge and the area scale diverges, so a cap of sky at the pole occupies an absurd share of the marks. Over a 70° cone that never shows.

Which is the general caution about every number in this essay. A spending ratio is quoted over a stated cone, and the surfaces differ in where their trouble lives — the plane’s is at the rim of any cone a reader picks, and the equirectangular surface’s is at a point that most cones exclude. Comparing two surfaces over a field that contains one’s difficulty and not the other’s is the same error as comparing them at a fixed focal length, one level subtler.

The honest summary is that the ranking above is a ranking over a 70° cone about the axis, it is stable in order across the fields this family draws, and the only surface whose position is an accident of the cone is the equirectangular one.

What this does not settle #

Three limits, and the third is the one that would change the numbers.

The comparison is over a cone of directions, symmetric about the axis, which is the right shape for a fisheye and the wrong shape for a photograph. Real pictures are rectangular, and a rectangular crop of a plane samples the surface’s worst corners while a rectangular crop of a fisheye throws them away. Nothing here accounts for the frame’s shape, and a comparison that did would narrow the gap.

The marks are treated as an area, not as a sampling lattice. A real sensor has a grid with an orientation, and the anisotropy the equidistant fisheye shows tangentially interacts with that grid in ways that are about sampling rather than about geometry. This collection computes the geometry.

And the whole comparison is about where the marks go, not about what reaches them. A wide lens has falloff, and the corners of a rectilinear picture are darker as well as more finely sampled. That is radiometry and it belongs to another subject; the geometry says only how many marks the corner has.

One budget, six ways of spending it #

The object is a picture surface, and the finding is that the question everybody asks of one — what does it distort? — is the second question rather than the first.

The first is where its marks go, and the six named surfaces answer it over a range of twenty-five to one at a field a real camera reaches. The surface with the best answer to the fidelity question has the worst answer to this one, and the surface whose reputation rests on being linear turns out to be linear in one of the two directions it has.