A pole is a line

An equirectangular picture puts azimuth across and elevation down, which is the simplest arrangement anybody could choose and is why almost every panorama is stored that way. It has one consequence that a reader meets the first time they look at the top of such a file and never quite gets an explanation of.

The zenith is not a point in the picture. It is the entire top edge. Every azimuth meets the pole, so every column of the image ends at the same direction, and one direction has been given a whole line of marks.

That is not a bug in a file format. It is a property every one-chart map of the sphere has to have, and the interesting question is not whether a surface has such a direction but what currency it pays in — because the six named surfaces pay in visibly different ones.

The second cap is the control and it is doing real work. Without it a reader could reasonably think the ×9.2 was a statement about equirectangular pictures in general — that the whole surface is wasteful. It is not: away from the pole the same computation on the same surface returns slightly less than an even share, because the mean area scale over this picture is π/2 rather than one.

The divergence is in the shrinking, not in the size #

The natural next question is how bad it gets, and the answer has an unexpected direction to it.

Doubling the cap halved the penalty, and that is the whole shape of a singularity. A cap of angular radius β at the pole is drawn as a band the full width of the picture and β tall, so its image area falls only as β while its solid angle falls as β². The ratio therefore grows as 1/β without bound as the cap shrinks.

So the surface is not badly behaved at the pole in the way a reader first imagines — it does not have a large error there. It has an error that grows without limit as the thing being measured gets smaller, which means there is no scale at which the surface is merely somewhat wasteful. Ask about a small enough patch of sky and the answer is as large as anybody cares to name.

Every one-chart map has one, and they fail in different currencies #

The pole is easy to see because the equirectangular surface makes it a whole edge. The other surfaces have the same problem hidden in a different place, and comparing them is where the essay earns itself.

The two extremes of that list are the finding, and they are opposites rather than a range.

The equal-area fisheye pays nothing in area at its antipode. The exponent is 0.01, which is to say zero: the scale is 1.0000 all the way in. It has a singular direction — the antipode is still sent to a whole rim circle rather than to a point — but the currency it pays in is shape. A small circle there is drawn as an extremely thin annular sliver of exactly the right area.

Stereographic pays nothing in shape and everything in area. Its conformality holds to 3 × 10⁻⁸ right up to the antipode, and its area scale runs away as β⁻⁴ — the steepest of the five by a wide margin.

That is the conformal-or-equal-area trade appearing at the one direction where it cannot be dodged. Elsewhere on the sphere a surface can be a compromise between the two; at its singular direction it has to fail, and the only choice is which of the two things to fail at. Stereographic keeps every angle and the third column is area measure the two properties across the sphere; this is the same trade at the point where it becomes forced.

Why the failure cannot be designed away, only moved #

It is worth being precise about the claim, because “every surface has a singular direction” sounds like an empirical observation about six particular surfaces and is not.

A sphere is compact and has no boundary. A region of the plane that a picture occupies is not the same kind of object — cut a sphere anywhere and it opens, and the cut is where the trouble goes. There is no continuous one-to-one map from the whole sphere onto a plane region, so any surface that claims to hold every direction in one picture has at least one place where the map stops being one-to-one, and that place is where a direction acquires more than one mark or a whole set of them.

The three azimuthal surfaces put it at the antipode of their axis and send that one direction to their entire rim circle. The equirectangular surface puts it at both poles and sends each to an edge. In every case the singular set is a curve in the picture standing for a point in the world, which is where the essay’s title comes from: a pole is a line, and it is a line because it has to be somewhere and a line is the cheapest place to put it.

What a surface designer can choose is where. What nobody can choose is whether. That is a statement about topology rather than about projection, and it is the one result in this collection that does not depend on a camera at all — it would hold for any way of drawing the sphere on a page, including ones nobody has thought of.

The cube map’s escape is genuine and it is the only escape: stop claiming to be one picture.

The worst place is not the one the folklore names #

Ranking the five by what a small cap actually costs turns up a surface nobody warns about.

The equidistant fisheye is the surface a reader is least likely to be suspicious of. It is the standard scientific fisheye, it is linear in angle, and its selling point is evenness. Every fisheye is a different rule sets out the four laws; nothing in that comparison suggests this one has the worst singularity of the group, and it does — for a reason that is purely about how far its rim is from its centre.

That is worth stating as a general caution. A surface’s behaviour at its singular direction is not predictable from its behaviour in the middle of the frame, and the properties that are quoted about these surfaces are all middle-of-the-frame properties.

The one that is bounded, and what it costs to be #

One entry in that ranking is different in kind, and it is the reason the essay’s title is a claim about one-chart maps rather than about maps.

The cube map has no singular direction because it never tries to hold the whole sphere in one chart. Six charts, each covering a comfortable 90°, each of them an ordinary plane well away from its own trouble — and a plane’s own trouble is at 90° off its axis, which is exactly where the lines a surface leaves alone shows the rectilinear picture running to infinity. The worst thing that happens anywhere is a corner at ×2.04.

The cost is that it is not one picture. A great circle crossing a seam is two straight segments meeting at an angle rather than a single curve — the kink at a seam measures what that costs — and nothing about the six faces makes a continuous surface. So the trade is exact and it is a trade about charts rather than about distortion: hold the whole sphere in one piece and something must go wrong somewhere, or cover it in pieces and pay at the joins.

That is a theorem rather than an engineering observation, and it is why the essay’s claim is unconditional. A sphere is not homeomorphic to a plane region; no single chart covers it; and the direction a chart cannot hold is where the failure has to appear.

What a small circle becomes there #

The area ratio is one reading of the singularity and it hides something a reader should know, which is what a shape does as it approaches.

At the equirectangular pole a small circle of sky is drawn as a band — very wide, very short. Its area is too large by the factor measured above, and its aspect is worse: the tangential scale grows without bound while the radial scale stays finite, so a circle becomes an ellipse of unbounded eccentricity. Approaching the pole, a round thing is drawn flatter and flatter until it is a line across the whole picture.

That is why the two currencies in the divergence figure are the right way to read it. The equal-area fisheye’s antipode has an area exponent of zero and is not well behaved there — its shape distortion is unbounded instead, and a circle at its antipode becomes a thin annular sliver of exactly the right area. Nothing is preserved at a singular direction; the surface only chooses which of the two accounts stays balanced.

Conformal is not undistorted makes the neighbouring point about the little planet, where a surface preserving every angle produces a picture nobody would call faithful. The pole is that argument at its limit: preserving angles exactly, all the way to a direction whose neighbourhood is magnified by a fourth power, is preservation of a kind that stops being useful.

Where a singularity is the point rather than the price #

One use of these surfaces inverts the whole essay and is worth naming, because it shows the singularity is a tool as well as a cost.

The little-planet picture is a stereographic projection taken with its singular direction pointed at the zenith, so the ground curls into a disc and the sky is stretched around the outside. Everything this essay calls a failure is what makes that picture work: the β⁻⁴ area divergence is what turns the sky into a wide, soft surround, and the exact conformality is what keeps the buildings on the little planet looking like buildings rather than smears.

So the same surface, with the same singularity, is a defect for storage and the entire point for a picture. What decides is where the singular direction is pointed and whether anything interesting is near it — which is the practical form of the choice the section below is about, and the reason a surface cannot be ranked without saying what it is for.

What a reader should take from it #

Three practical consequences, each following directly from a number above.

A surface’s worst place should be pointed at nothing. That is the actionable form of the whole essay, and it is a decision made when the rig is set up rather than in software afterwards. What a 360 photograph is describes the object; where its singular direction ends up is a choice, and the only bad choice is putting it somewhere a reader will look.

Storage is not neutral. An equirectangular file spends 4.50 per cent of its marks on 0.49 per cent of the sky, and the imbalance grows as the patch shrinks. That is why so much machinery exists to re-tile panoramas before working with them, and why the cube map is the format of choice for anything that has to sample the sphere evenly.

A resolution quoted for a panorama is a resolution somewhere. “Eight thousand pixels across” describes the equator. Near the pole those same columns are describing one direction, and a feature there has far fewer independent marks on it than the number suggests. Where a surface spends its pixels makes the same point across the middle of the frame; this is its extreme.

And the choice of singular direction is a real decision. Every azimuthal surface puts its trouble at the antipode of the axis, which for a sky camera pointed up is the ground — a direction usually occupied by a tripod and of no interest. The equirectangular surface puts it at the zenith, which for a landscape panorama is empty sky and for an interior is the ceiling. Choosing the surface is partly choosing where to put the place that does not work.

The singular direction and the reach are the same fact #

One more connection is worth drawing, because it ties this essay to the plainest property a surface has and shows the two are not independent.

A surface’s reach is how far off its axis it can put a mark at all. The plane reaches strictly less than 90°; the equidistant, equal-area and stereographic fisheyes reach a full half-turn or beyond; the equirectangular surface reaches everything. And the singular direction is always at the edge of the reach — it is the direction the surface is straining to include, drawn at the boundary of what it can hold.

So the plane, which has the worst spending anywhere in the middle of the frame, has no singular direction at all in the sense of this essay: it does not reach 90°, so the directions that would be singular are simply absent. Its area scale runs to infinity as the field approaches a half-turn, which is the same divergence seen from the other side — it appears as an unreachable boundary rather than as a compressed edge. Where a surface spends its pixels measures that from inside the reachable region.

That gives a clean way to state what the six surfaces have done with one unavoidable problem. Either the surface declines to reach the difficult directions, as the plane does, and pays by covering less than half the sphere. Or it reaches them and compresses them onto a curve, as the four wide surfaces do, and pays in area or in shape at that curve. Or it declines to be one picture, as the cube map does, and pays at the seams.

Three strategies, and no fourth, because there are only three things to do with a topological obstruction: avoid it, absorb it, or cut around it.

Which of the three a surface takes is the first thing to know about it, and it is not what any of these surfaces is usually described by. The familiar descriptions — conformal, equal-area, linear in angle — are all statements about the interior of the reachable region, and they say nothing at all about the strategy. Two surfaces with the same interior behaviour can take different strategies, and two with the same strategy can behave quite differently inside.

What this does not settle #

The measurement is of the map, not of any sampling of it. A real file has pixels, and the interaction between a diverging area scale and a finite grid is an aliasing question rather than a geometric one; this collection computes the geometry, and the divergence is in the map before any pixel exists.

Nor does anything here say a panorama looks wrong at its pole. Most viewers reproject before displaying, so the storage surface’s singularity is never seen — which is exactly why it is worth measuring rather than looking at. The waste is real, it is in the file, and no amount of looking at a rendered view reveals it.

And the exponents are fitted over a stated range of β and reported as fitted. The equal-area fisheye’s 0.01 is a measurement of zero rather than an assertion of it, which is the distinction this collection tries to keep: a construction that asserts even area returning an exponent of 0.01 is the arithmetic agreeing, and a construction that asserted it and returned 0.5 would be a bug found.

One direction, and what it costs to draw it #

The object is a single direction — the one a chart cannot hold — and the finding is that every one-chart surface has one, that they differ by a factor of nearly twenty in what a small cap costs there, and that the two surfaces with a zero in the comparison have their zeros in different columns.

The equal-area fisheye pays nothing in area and everything in shape. Stereographic pays nothing in shape and everything in area, at a fourth power. And the only bounded answer belongs to the surface that declined to be one picture at all.