Conformal is not undistorted

The word distorted does a lot of work in every discussion of wide pictures and it never gets a definition. A fisheye is called distorted; a wide flat photograph is called distorted at the edges; a panorama is called distorted at the top. In each case the word is standing in for a comparison the speaker has not made, and the comparison is impossible anyway, because there is nothing to compare a picture with except another picture.

This essay takes the one case where the word is used most confidently and shows that the picture in question is exact in the property the word usually means.

What the picture is #

A spherical panorama is stored as an equirectangular picture: azimuth across, elevation down, one rectangle addressing every direction. It preserves nothing at all, and it is not meant to — it is a way of addressing directions, chosen because the addressing is simple.

A little planet is that picture re-projected. Each pixel’s address is turned back into a direction, the sphere is rotated so that the nadir — straight down — comes to the centre, and the direction is sent through the stereographic map. The ground, which occupied the bottom row of the equirectangular rectangle, becomes a disc in the middle of the result; the sky, which occupied the top row, is smeared around the outside.

Written as a composition it is inverse ∘ rotate ∘ map, and no image data is involved anywhere. Every property measured below is a property of that composition, differenced on the sphere.

The measurement #

The site’s curved field measures conformality in two halves, and it insists on both because the second is the one that gets dropped.

Does a right angle stay a right angle? Measured by differencing the map along a pair of perpendicular tangent directions and taking the worst case over every rotation of that pair, rather than over the surface’s own coordinate directions. That distinction is not a nicety: differencing along the surface’s own axes gives the cylinder a perfect score, because tangentBasis happens to produce the cylinder’s azimuth and elevation and those two do stay perpendicular. Rotating the right angle through a half-turn rejects the cylinder by 5.62°.

Are the two arms magnified equally? Measured as the ratio of the Jacobian’s two singular values — the most and the least a direction can be magnified. This is the half that gets dropped, and dropping it calls a cylindrical panorama conformal, which it is not: a shape near the top of one is visibly taller than it should be.

Over 160° of the sphere the little planet’s answers are 4.4e-8° and 1.000000023. The first is the differencing step’s own noise. The second is one, to eight decimal places.

There is no direction in the picture in which a right angle is bent, and no direction in which one arm is magnified more than the other. Every crossing in the world crosses at the same angle in the picture, and every infinitesimal shape is similar to the shape it came from.

The half of conformality that gets dropped #

It is worth dwelling on the second half of the test, because the first version of this site’s machinery failed it and the failure is instructive.

Conformality is usually stated as “angles are preserved”, and a natural implementation differences the map along two perpendicular tangent directions and reports the angle between the results. That implementation reported 5.5e-10 degrees for the cylinder — a perfect score, for a surface that is plainly not conformal, since a shape near the top of a cylindrical panorama is visibly taller than it should be.

The reason is that the tangent basis the code builds happens to align with the cylinder’s own azimuth and elevation, and those two do stay perpendicular. Every other right angle at the same point does not. Rotating the pair through a half-turn and keeping the worst case rejects the cylinder by 5.62°.

Both versions are kept in the library and the site’s gate asserts that the naive one passes and the honest one rejects, because a test evaluated at the one input where it cannot fail measures nothing. That is the second time this site has made that mistake in a different costume; the first was a cross-ratio evaluated over four consecutive divisions of a receding row.

The consequence for this essay is that its zero has to be read as the rotated zero. A little planet is conformal in the strong sense: every right angle at every point, in every orientation, and both arms magnified equally.

And what has gone #

The third measurement is where the picture’s reputation comes from.

The area scale — the Jacobian determinant, against its value on axis — runs from one at the centre to 255 times that at 160° off axis. Push the sample to 176° and the ratio is 4762.

So a person standing near the middle of a little planet and a person standing near its edge are drawn at exactly the same shape, in the strict sense that every angle in both is preserved, and at sizes differing by a factor that runs into the hundreds.

That is the whole of the effect. Nothing is bent; everything is resized. And the resizing is not uniform across the picture, so it cannot be undone by looking at it differently.

The claim about the word #

Put the three numbers together and the sentence writes itself.

The picture is conformal and unrecognisable, and those are not in tension. A reader calling it distorted is reacting to the third measurement and using a word that names the first two.

That is worth insisting on because the confusion has consequences beyond this one picture. “Conformal” is routinely offered as the property that makes a projection faithful, and it is offered by people who would not defend the claim if it were spelled out. A conformal map is faithful about angle and about nothing else. It has no obligation to area, to distance, to straightness, or to a viewer’s sense of what a room looks like, and this picture is the case where it discharges its obligation completely and satisfies nobody.

The peak is the square of the average #

The area figures quoted so far are peak-to-centre ratios, taken at the very rim of the sample. That is one statistic among several, and stereographic has a relation between two of them that is worth having, because the two differ by a square.

The map sends a direction $\theta$ off the centre to radius $2\tan(\theta/2)$, so a cap of angular radius $\Theta$ images as a disc of area $4\pi\tan^{2}(\Theta/2)$ while the cap itself subtends $4\pi\sin^{2}(\Theta/2)$. Divide:

$$\langle\text{area scale}\rangle \;=\; \sec^{2}\frac{\Theta}{2}, \qquad \text{peak area scale} \;=\; \sec^{4}\frac{\Theta}{2},$$

the second being the Jacobian at the rim. The peak is the square of the average, exactly, at every sample size. At 160° that is 33 against 1,100; at 176°, 821 against 674,000.

Which of the two a reader is reacting to is not obvious and the difference is enormous. The peak is attained on a circle of zero width — an infinitesimal ring at the very edge of the picture — while the average is what the picture as a whole spends. Quoting the peak is quoting a limit; quoting the average says how much of the sphere the typical pixel is carrying, and it is the smaller and more defensible number.

There is a third statistic and it is the one that makes the picture legible. Because image area goes as $\tan^{2}(\theta/2)$, the fraction of a little planet lying within $\theta$ of its centre is

$$\frac{\tan^{2}(\theta/2)}{\tan^{2}(\Theta/2)}.$$

On a 160° little planet the whole lower hemisphere — everything below the horizon, which is to say the ground the reader is standing on and the entire subject of the photograph — occupies $\tan^{2}45°/\tan^{2}80° = 3.1$ per cent of the picture’s area. The remaining 97 per cent is the last seventy degrees of sky. That is the complaint stated as a budget rather than as a distortion, and it explains why the effect reads as a planet: the ground is compressed into a small central disc not because anything is bent but because the surface has spent almost all of its paper on the sky.

Two consequences for the argument the essay is making.

The three numbers are not independent. Conformality forces the two magnifications equal, so the area scale is the square of the single linear scale, and the whole of the area behaviour is determined once the rule is fixed — which is why only one rule is conformal and why its area behaviour is not a design choice that could have gone another way. Asking for a conformal picture is asking for this area budget.

And the honest measure has to name its sample. The peak ratio depends on $\Theta$ as a fourth power, so a little planet made from 160° of sphere and one made from 176° differ by six hundred times in the statistic and by twenty-five in the average. A number quoted without the field it was taken over is not a property of the surface, which is the same care the surface table takes in stating its own comparisons at a fixed 80° and the cloud count takes in holding the counted field the same for every lens.

The measurement that would have said what a reader means #

If the reader’s word means something, some measurement captures it, and the honest thing is to say which.

It is not the angular error, which is zero. It is not the anisotropy, which is one. It is the range of the area scale across the picture — the ratio of the largest local magnification to the smallest — and this site can report it for every surface it has.

That measurement gives an ordering that matches the ordinary complaint. A narrow flat picture: near one. A wide flat picture: several. A fisheye of the equal-area kind: exactly one by construction, which is the point of it. A little planet at 160°: 255.

So “distorted”, used the way readers use it, is approximately “the area scale varies a lot across this picture”, and it is a property no discussion of conformality touches. Naming it lets the complaint be answered rather than dismissed.

What an equal-area picture would look like instead #

The natural response to all this is to ask for the other side of the trade, and the site has it: an equal-area fisheye, in which equal solid angles image to equal areas by construction.

Its area scale is exactly one everywhere — the site measures it as an exact zero, one of only two in the whole surface comparison — so on the measurement a reader is reacting to it is perfect. And it is the wrong surface for reading a shape, which is why it is used for counting things in a sky and not for looking at rooms.

Its angular error over the same field is large, its anisotropy is well away from one, and the practical form of that is that a face near the edge of an equal-area picture is the right size and the wrong shape. The complaint changes rather than going away, and it changes into one readers voice less often because a squashed face at the correct size reads as a squashed face and a correct face at the wrong size reads as a distorted picture.

That asymmetry in what gets complained about is worth naming, since it is the reason the reader’s word attached itself to area in the first place. It is not evidence that area matters more; it is evidence that size errors are easier to notice than shape errors when nothing in the picture states a scale.

The trade cannot be escaped #

There is no surface that keeps angle and area at once, and the reason is short enough to state.

A map preserving angle multiplies every direction by the same local factor; a map preserving area has a Jacobian determinant of one. A map doing both has a local factor whose square is one everywhere, which makes it an isometry — and there is no isometry from a sphere to a plane, because their curvatures differ. That is Gauss’s theorem, and it is the same shape of impossibility as Beltrami’s, which forbids being straight and conformal together.

So the three properties this field measures are not three independent goods that a clever surface might get. They come in pairs that exclude each other, and every picture surface is a decision about which pair to break.

Why the little planet reads as a planet #

There is one more thing the measurements say, and it explains the picture’s name rather than its reputation.

Stereographic sends circles to circles. The horizon of the original panorama is a circle on the sphere, so it is a circle in the picture; every level line on the ground is a circle too. The result is an image whose ground is bounded by a circle with everything inside it, which is what a small spherical body photographed from above would look like — and a viewer reading it as a planet is reading a correct consequence of the map rather than making a mistake.

The sky’s smear around the outside is the same circle-preservation seen from the other side: the zenith, a single point, has been sent to infinity, and the directions near it are drawn very large.

The tolerance the measurement needs #

One technical note, because it is the sort of detail that turns a zero into a fiction.

The angular error is quoted at 1e-6 rather than at 1e-9, and the difference is about the map rather than about the code. Stereographic’s derivative grows without bound toward the antipode, so a central difference at $10^{-5}$ radians is subtracting two large nearly-equal numbers by 176° off axis, and it gives back about seven digits instead of fifteen.

Tightening the tolerance would not measure the surface better. It would measure the step size, which is exactly the failure the field’s own control — differencing the flat plane against its closed form $\sec^3\theta$ — exists to catch. Quoting a zero to more digits than the arithmetic supports is the same error as quoting a residual in absolute units, and both are ways of reporting the instrument.

The same confusion one field away #

The distinction this essay draws is not local to picture surfaces, and the site has the neighbouring case already built.

The parallel field’s ruler essay measures the anisotropy of an isometric drawing’s coordinate plane and finds $\sqrt3$ — a unit segment is drawn anywhere between 0.5774 and 1.0000 depending on its direction. That is a shape failure with the area under control, and the practical consequence is that a ruler on the paper is up to 29.3% short and up to 22.5% long.

Nobody calls an isometric drawing distorted. It has a shape error of the kind this essay’s picture does not have, and it escapes the word because its area scale is even and its lines are straight.

Two drawings, then: one exact in shape and wrong in size by a factor of hundreds, universally called distorted; one exact in size and wrong in shape by a factor of 1.7321, universally called a technical drawing. The word is tracking one of the three measurements and it is not the one the mathematics names.

The short version #

A little planet is a spherical panorama re-projected stereographically from below. Its worst angular error over 160° of the sphere is 4.4e-8°, its worst anisotropy is 1.000000023, and its area scale runs over a factor of 255 — rising to 4762 if the sample is pushed to 176°.

So the most distorted-looking picture in circulation is exact in angle and shape, everywhere, and wrong only in size. The word readers reach for names the property it has and not the property it lacks.

The measurement that captures the complaint is the range of the area scale, and it is available for every surface. Using it instead would let the objection be answered with a number rather than argued about with a word.