Visual Plane
every picture states the point it is correct from
Surfaces that are not flat

Conformal is not undistorted

The most distorted-looking picture in ordinary circulation is the little planet — a 360 photograph re-projected from below, with the ground curled into a ball. Its worst angular error over 160 degrees of the sphere is 4.4e-8 degrees, which is arithmetic noise. Every crossing in the original crosses at exactly the same angle in the result, and what has gone is area, over a factor of 255.
18 min read 12 figures The surface is a choiceWhat a projection destroysTaught, and never measured

Worth reading first: Stereographic keeps every angle, and only stereographic · What a 360-degree photograph actually is · When the picture surface is not flat.

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.

Every angle exact, and unrecognisableA 360 photograph re-projected stereographically from below. Every crossing in the original crosses at the same angle here — the worst departure over 160° of the sphere is 4.4e-8° — while the area scale runs over a factor of 255. Conformal is not a synonym for undistorted.angle, worst over the sphere4.4e-8°anisotropy, worst1.000000023area scale, largest over smallest×255what a reader calls distortedthe third row, not the firstthe disc is 160° of the spheredrawn to 160° off axisthe first two rows are conformality
Fig. 1 A 360 photograph re-projected stereographically from below, with the grid of the sphere drawn on it. The worst angular error over 160° of the sphere is 4.4e-8°, the worst anisotropy is 1.000000023, and the area scale runs over a factor of 255. The first two numbers are what conformal means and the third is what a reader is reacting to.

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.

Every angle exact, and unrecognisableA 360 photograph re-projected stereographically from below. Every crossing in the original crosses at the same angle here — the worst departure over 100° of the sphere is 3.9e-9° — while the area scale runs over a factor of 5. Conformal is not a synonym for undistorted.angle, worst over the sphere3.9e-9°anisotropy, worst1.000000024area scale, largest over smallest×5what a reader calls distortedthe third row, not the firstthe disc is 100° of the spheredrawn to 100° off axisthe first two rows are conformality
Fig. 2 The same re-projection sampled over less of the sphere. The angles are still exact and the area range is only ×5, which is the honest form of the claim: the runaway is a property of how much sphere is included rather than of the surface alone.

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.

Every angle exact, and unrecognisableA 360 photograph re-projected stereographically from below. Every crossing in the original crosses at the same angle here — the worst departure over 176° of the sphere is 2.0e-7° — while the area scale runs over a factor of 4762. Conformal is not a synonym for undistorted.angle, worst over the sphere2.0e-7°anisotropy, worst1.000000025area scale, largest over smallest×4762what a reader calls distortedthe third row, not the firstthe disc is 176° of the spheredrawn to 176° off axisthe first two rows are conformality
Fig. 3 The extreme sample, reaching nearly to the antipode. The angular error is still under a microdegree and the area range is ×4762, which is the runaway with nothing left to hide it.

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.

Straight, or conformal — the corner that is emptySix picture surfaces plotted by how much they bend a straight line (across) against how far they are from preserving shape (up). The plane sits on the left edge and the top; stereographic on the bottom and the right. Nothing sits in the bottom-left corner, and by Beltrami's theorem nothing can.straight AND conformal — empty10⁻⁷10⁻⁶10⁻⁵10⁻⁴10⁻³10⁻²10⁻¹10⁰10⁻⁷10⁻⁶10⁻⁵10⁻⁴10⁻³10⁻²10⁻¹10⁰10¹10²bend of a straight line, as a fraction of its own chorddeparture from conformal (degrees, or % of anisotropy)planecylinderstereographicequidistantequal-areaequirect.sampled over a 130° fanlower left would be a surface with no cost
Fig. 4 The trade, over the whole family. Stereographic sits alone at the conformal edge and the plane alone at the straight edge; nothing occupies the corner where a surface would be both, and no surface anywhere is faithful about everything.

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.

Area scale across the picture, for six surfacesThe flat plane's area scale is sec³θ and leaves the frame before 70°. The equal-area fisheye is the flat line at 1, which is what its name asserts and what this measures.024680204060angle off the optical axis (degrees)area scale, relative to the centreplanecylinderstereographicequidistantequal-areaequirect.measured by differencing on the spherethe plane leaves the frame
Fig. 5 The area scale across the field for the fixed six. The equal-area fisheye’s curve is flat by construction and the stereographic one climbs fastest, which is exactly the ordering the ordinary complaint produces.

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.

One room at 130° across, cast onto six picture surfacesEach panel holds the same angular width of the same scene. The flat plane keeps every straight line straight and pays for it at the edges; the five curved surfaces bend the ground lines by 3.5% to 6.0% of their own length.plane — bend 0.0%cylinder — bend 6.0%stereographic — bend 3.5%equidistant — bend 4.5%equal-area — bend 5.1%equirect. — bend 5.7%130° across in every panelsame scene, same angle, six surfaces
Fig. 6 The six decisions, side by side. Each panel is a different answer to the same impossibility and none of them is the corrected version of another.
Five great circles, imaged on the stereographicEach is fitted as a general conic and comes back a circle: |A−C|+|B| is 1e-9 of the fit's own scale. Stereographic is the only surface here that does this.fitted conic: a circle to 1e-9stereographicdashed: the fit, not the samples
Fig. 7 The property that gets stereographic its exactness: every circle on the sphere images as a circle, which is stronger than conformality and implies it. That is also why a little planet’s horizon is a perfect circle rather than an approximate one.

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.

Six flat pictures, and what happens where two of them meetEach face is a flat picture at 90°, so a straight line inside one is drawn exactly straight — 1e-15 of its chord. Across a seam the two straight pieces meet at 1.80°. The shading is the area scale, which runs from 1 at a face's centre to 5.196 at its corner, with an anisotropy of 1.7321 there.leftfrontrightbackupdownacross the left/front seam: 1.80°, with each side straight to 7e-16corner area ×5.196anisotropy √3 = 1.7321 there
Fig. 8 For contrast, the answer that keeps straightness and gives up continuity. Its area scale runs over a factor of only 5.196 from face centre to corner, so it is far less “distorted” in the reader’s sense and far worse in every other.
One family, one parameter, and where the convention sits in itThe same room on five members of the Panini family, each scaled so 130° of the world spans the same width. Every member keeps a vertical line vertical; only d = 0, the flat plane, keeps a general one straight. The worst angular error over the field is smallest at d = 1.00, at 16.84°.d = 0d = 0.5d = 1d = 2d = 4d = 0straight48.06° angle×17.80 aread = 0.54.81% bend23.08° angle×4.66 aread = 17.21% bend16.84° angle×2.65 aread = 29.61% bend23.13° angle×1.57 aread = 411.54% bend32.71° angle×1.73 areaa general straight line · worst angle · area range, over the field130° acrossangular minimum at d = 1.00
Fig. 9 And the compromise family, which declines both extremes. Its area range at the conventional setting is ×2.65 over a 130° field, against the little planet’s ×255 over 160° — two orders of magnitude apart, on the measurement a reader is actually reacting to.
The lines each surface leaves aloneA curved picture surface does not bend everything. Each panel draws the family of world lines the surface images as straight lines: two-dimensional for the plane, and a one-parameter family for every other surface here — running through the picture's centre on an azimuthal surface, and parallel on a cylindrical one.planeevery line—cylinderone parameter · parallelno meeting pointstereographicone parameter · concurrentthey meet to 1e-12equirect.one parameter · parallelno meeting point3 of 4 keep a curvethe signature has three values, not eight
Fig. 10 The lines each surface draws straight. Stereographic’s are the radials through the picture’s centre, so a little planet has a family of exactly straight lines running out from its middle — and nothing else in it is straight at all.

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 10−510^{-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⁡3θ\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 3\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 image of a circle in the xy plane, in 4 systemscavalier draws this plane isotropically — the circle stays a circle at 1.0000, so a length in it can be taken off the paper with a ruler whichever way it runs. The others draw it as an ellipse, and the ratio printed under each is the factor a ruler is wrong by between the best direction and the worst.isometric0.57741 : 1.732dimetric0.88191 : 1.134cavalier1.0000a circlemilitary0.57741 : 1.732the xy plane's drawn ellipseratio of the ellipse's axes, sampled
Fig. 11 The parallel field’s version of the same three-way trade. A unit circle drawn in four systems: the ellipse is the set of drawn lengths, and its eccentricity is the anisotropy — a shape error of exactly the kind a conformal map does not have.
Every angle exact, and unrecognisableA 360 photograph re-projected stereographically from below. Every crossing in the original crosses at the same angle here — the worst departure over 160° of the sphere is 4.4e-8° — while the area scale runs over a factor of 255. Conformal is not a synonym for undistorted.angle, worst over the sphere4.4e-8°anisotropy, worst1.000000023area scale, largest over smallest×255what a reader calls distortedthe third row, not the firstthe disc is 160° of the spheredrawn to 160° off axisthe first two rows are conformality
Fig. 12 And the picture this essay is about, once more with the three numbers in view. Two of them are zeros and one of them is 255, and the word attaches to the third.

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.

Where this goes next

fitted conic: a circle to 1e-9stereographicdashed: the fit, not the samples Stereographic keeps every angle, and only stereographic One surface in the family preserves shape exactly — every right angle stays a right angle and both its arms are magnified equally, to the last bit the arithmetic has. It also sends every circle in the world to a circle in the picture, which the site's existing conic fit can be pointed at and asked to confirm without being told what it is looking at. no picture at 300°the plane is unbounded at 180°plane — bend 0.0%cylinder — bend 6.0%stereographic — bend 3.5%equidistant — bend 4.5%equal-area — bend 5.1%equirect. — bend 5.7%300° across in every panelsame scene, same angle, six surfaces What a 360-degree photograph actually is The format every spherical camera writes preserves nothing — not straightness, not shape, not area — and it is the right choice anyway, for a reason that has nothing to do with looking at it. An equirectangular file is a lookup table of directions, and the picture only exists at the moment something re-projects a piece of it. plane — bend 0.0%cylinder — bend 6.0%stereographic — bend 3.5%equidistant — bend 4.5%equal-area — bend 5.1%equirect. — bend 5.7%120° across in every panelsame scene, same angle, six surfaces No picture surface keeps everything A picture has to be cast onto something, and every candidate surface destroys something different. Six of them are measured here on the same three properties, and the corner of the plot where a surface pays nothing is empty — not because nobody has thought of one, but because a theorem says there is none.

Shares its objects with

Essays that name at least two of the same things, and that neither author linked.

Named objects

A flat tag is an object no other essay names yet.

AnisotropyArea scaleCircle preservingConformalDemonstrationEqual area projectionEquirectangularfield of viewJacobianLittle planetMarginal distortionnecessary, not sufficientPicture surfaceSpherical panoramaStereographic projection