The third column is area

Every picture surface here has been put through the same two questions. Does it keep straight lines straight? Does it keep shapes? The answers make a table with a hole in one corner — nothing keeps both, and Beltrami’s theorem says nothing can.

Both of those are questions asked by somebody looking at the picture. There is a third question, asked by somebody counting in it, and it has been answerable from the same machinery since the day the machinery was written. Nobody asked it.

Counting is not a marginal use. How much of the sky is cloud, how much of the canopy is leaf, how much of a wall a projector covers, how much of a hemisphere a lamp reaches: every one of those is a fraction of solid angle, and every one is measured by counting picture. Whether that is legitimate depends on a property nothing in this field had measured.

What area scale is #

At a direction, a picture surface maps a small patch of the sphere to a small patch of the picture. The ratio of the two areas is the area scale, and it is the determinant of the map’s differential — the same object whose two singular values give the anisotropy this field already reports.

So it costs nothing new. The differencing that was written to measure whether a right angle survives, and by how much the two arms are magnified, produces the determinant on the way past. It was returned, stored, and never plotted.

Normalising by the value on the axis is what makes it a claim about the surface rather than about the picture’s size. A picture can be printed at any scale; what matters is whether a square degree at the edge of the frame prints the same area as one in the middle.

The trap the cylinder set, again #

The first version of this measurement swept straight across the field — directions at increasing azimuth, elevation zero — and reported the cylinder’s area scale as exactly 1.000 at every angle.

That is wrong, and it is wrong for a reason this field has already recorded once.

A cylinder about the vertical preserves area along its own equator. Sweeping straight across runs along that equator, so the sweep is evaluated at exactly the directions where the surface cannot fail. The same file already contains a measurement that fails this way on purpose: the naive conformality test differences along the surface’s own tangent basis, gives the cylinder a perfect score, and is kept because the failure is the essay.

Walking into it a second time, by a different function, some rounds of work later, is worth recording rather than quietly fixing. A test evaluated along a surface’s own axis is a test evaluated where it cannot fail, and the shape of that mistake is not memorable enough to be avoided by remembering it.

The sweep here runs at 45° to the axes, and the cylinder reaches 2.71 times.

The family, measured #

Out to 80° off axis, area scale relative to the axis:

The first row is the result and the last row is the warning.

The equal-area fisheye is exact, to $8\times10^{-8}$, which is the differencing noise. That is not an approximation that happens to be good; the rule $\rho = 2\sin(\theta/2)$ is constructed so that the Jacobian determinant is constant, and the measurement confirms it rather than discovering it. What is worth having is the confirmation, because the name is often attached to lenses that do not obey the rule.

The flat plane is catastrophic. Its area scale is $\sec^3\theta$, so it grows without bound as the field approaches 180°, and by 80° a square degree at the edge prints 191 times the area of one on the axis. Counting pixels in a wide rectilinear photograph is not approximately measuring solid angle; it is measuring something else entirely.

And the two in between are not small. A 42% error at the edge of an equidistant fisheye, or a factor of three on stereographic, would ruin any count they were used for, and neither is a surface anybody thinks of as extreme.

What the numbers mean on a real frame #

The table is in ratios and a reader wants to know what they cost, so here are three readings of the same fact.

A rectilinear photograph at 90° across. Its corners are about 55° off axis, where the area scale is roughly 5.4. So a cloud in the corner of the frame occupies about five and a half times the picture area of an identical cloud in the middle. Anybody estimating cover by eye from such a frame is over-weighting the corners by that factor, and anybody counting pixels is doing it exactly.

A fisheye sold as “equidistant”, used for canopy cover. Its rim is 90° off axis; by 80° it is 1.42 times, and the error is one-sided — the edge of the frame is over-represented, so an obstruction near the horizon counts for more than one overhead. For a canopy measurement, where the horizon ring is exactly where the branches are, that bias runs the wrong way and is not small.

And a stereographic reprojection, which people reach for because it looks best. Out to 80° it is 2.9 times, and taken to the extremes a little-planet projection uses it runs to a factor of over a thousand. It is the correct choice for shape and the worst available choice for counting, and the two facts are the same fact: conformality and equal area cannot be had together.

The general form of the caution is that the error is systematic and one-directional. It does not average out over many frames, it does not shrink with a better sensor, and it does not announce itself, because a picture on the wrong surface is a perfectly good picture.

Why it has to be given up #

The obvious next question is whether some surface could keep straight lines, shape and area at once. It cannot, and the reason is the theorem the neighbouring rung is about.

A surface preserving both angles and areas everywhere is an isometry, and an isometry from the sphere to a plane would mean the sphere has zero Gaussian curvature. It does not. So conformal-and-equal-area is impossible before any candidate is examined, which is why stereographic — the conformal one — has the second-worst area behaviour in the table, and the equal-area one bends every line it touches.

That makes three properties and a set of exclusions rather than a ranking, and the table is the honest form of the field’s whole position: perspective is a row rather than a header, and so is every other surface.

What each surface is for #

Once there are three columns, the surfaces stop being a ranking and become a set of tools, each with a job it is exactly right for.

Flat plane — when straightness matters and the field is narrow. Architecture, documents, anything where an edge must stay an edge. Never for counting.

Stereographic — when local shape matters. A face at the edge of a wide picture is the right shape on stereographic and squashed on everything else. Never for counting.

Equidistant — when angular distance from the axis is what is being read. A surveyor measuring the elevation of a horizon obstruction wants radius proportional to angle, and gets it. Not for counting, and not badly wrong for it either.

Equal-area — when a fraction of solid angle is what is wanted, and only then. It bends everything and it is the only member of the family whose counts are right.

Cylinder — when the field is wide horizontally and narrow vertically, and verticals must stay vertical. Panoramas. Its area behaviour is poor at high elevations and nobody photographs the zenith with one.

Where the area goes on each surface #

It is worth knowing not only how much a surface distorts area but where, because the two surfaces with similar worst-case numbers put their error in completely different places.

The flat plane puts all of it at the edges. Its area scale is $\sec^3\theta$, which is nearly 1 out to 30° and then climbs steeply. A rectilinear frame is trustworthy for counting in its middle and useless at its corners, and there is no gradual region between.

The cylinder puts it at high elevation and none at all across. Along the horizon its area scale is exactly 1 — which is the trap this essay opened with — and above and below it climbs. So a cylindrical panorama counts a horizontal band correctly and a zenith badly.

Stereographic puts it everywhere, smoothly. It has no flat region: the area scale departs from 1 immediately and keeps going, which is the price of holding every angle exactly everywhere.

And the equal-area fisheye puts it nowhere, which is the definition, at the cost of bending every line that is not a radius.

That distribution matters more than the worst case for most jobs. A measurement confined to a narrow cone about the axis can be made on almost any surface; one that uses the whole frame needs the surface whose error is flat rather than the one whose worst case is smallest.

The measurement’s controls #

A table where one row is exactly 1 and another is 191 invites the suspicion that the machinery is doing something different in the two cases, so it is worth saying what keeps it honest.

One code path. Every row comes from the same central differencing on the sphere. There is no closed form used anywhere in the table.

One closed-form check. The flat plane’s area scale is $\sec^3\theta$, which is short enough to write down, and it is the only place in this field where the differencing can be compared against something that did not come out of it. It agrees. If that drifted, every number in the table would be reporting the step size.

And a sweep that is not along anybody’s axis. The 45° bearing, for the reason above.

The equal-area row’s $10^{-8}$ is therefore differencing noise rather than an assertion, and the 191 is a measurement rather than a formula quoted.

Reading the rule off a photograph #

If area scale depends on the rule, and a lens is sold with a name rather than a rule, the practical question is how to find out which rule a given lens obeys. It is answerable from a photograph, and it is the same trade the rest of this site makes.

Photograph something with known angular structure. A row of marks at known angles from the axis — a calibrated target, or a set of stars, or a plumb line at a measured distance — gives pairs of (angle, radius in the picture).

Fit the radius against the four candidate rules, each with one free scale, because the size of the picture is not part of the claim. The rules are $\rho \propto \theta$, $2\sin(\theta/2)$, $2\tan(\theta/2)$ and $\sin\theta$, and they separate at large angles even when they agree near the axis.

And look at the residual, not the best name. All four agree to first order about the axis, so a fit over a narrow field identifies nothing. What identifies a lens is the shape of the curve at 60° and beyond, and a residual of a per cent or two of the picture’s radius is the difference between a lens that obeys a rule and one that merely resembles it.

That procedure has a pleasant consequence which belongs to another field on this site: a mirror ball, photographed from far enough away, turns out to obey the equal-area rule exactly, and it does so because of a two-line argument about reflection rather than because anybody designed it that way.

The third column changes what a picture is for #

The habit worth taking from this is not about lenses.

A picture surface is a measuring instrument with a stated calibration, and which calibration is the right one depends on the question. There is no undistorted picture, so there is no default. Choosing a rectilinear lens because it looks natural and then counting pixels in it is choosing an instrument for its appearance and using it for a measurement it does not make.

That framing also says when to correct rather than re-shoot. A count made on the wrong surface is recoverable — the area scale is known, so each pixel can be weighted by it — provided the rule is known. What is not recoverable is a count made on an unknown surface, and “fisheye” is not a rule.