Stereographic keeps every angle, and only stereographic
Of the six picture surfaces this site computes, five distort shape and one does not. That sentence is easy to write and hard to earn, because shape is the property whose informal version and formal version part company, and almost every claim made about it in photography is made in the informal one.
The formal version has two halves. A surface preserves shape at a point when a right angle there is still a right angle in the picture, and when the two arms of that angle are magnified by the same amount. Keep the first and drop the second and a square becomes a rectangle, which is a change of shape by any reasonable use of the word. Both halves together are called conformality, and one surface here has it.
What conformal means, checked both ways
The stereographic map takes a direction with components (x, y, z) and returns the picture point
which is the classical projection from the far pole of the sphere onto the tangent plane at the near one. It has been known to be angle-preserving since Ptolemy used it for star charts, and the standard demonstration is a page of algebra.
The check here is not the algebra. It differences the map on the sphere: at a direction, two perpendicular tangent vectors are pushed through, and the angle between the results is compared with ninety degrees. Then the pair is rotated through a half-revolution in ninety steps and the worst case kept, because a pair that happens to align with a surface’s own coordinate directions can survive when nothing else does — that is exactly how a first version of this measurement gave the cylinder a perfect score.
Stereographic’s worst departure over a fan of forty-five directions spanning 120° across and 50° up is 1.7 × 10⁻⁹ degrees. The anisotropy — the ratio of the largest to the smallest magnification, read off the singular values of the map’s derivative — is 1.000000022. Both are the noise floor of central differencing at a step of 10⁻⁵ radians, which is to say both are zero.
Every other surface fails one half or the other, and the nearest miss is instructive. The equirectangular surface’s angle error over the same fan is small, and its anisotropy is 1.103 — a ten per cent stretch, invisible in the angle test and plainly not a preserved shape. Reporting only the angle would put it beside stereographic, and it is nowhere near it.
Why the projection has to be from the far side
The formula above is compact enough to look arbitrary, and the construction behind it is short enough to be worth having, because the construction is where the angle-preserving property comes from.
Put the sphere of directions with its near pole touching the picture plane. To find the mark for a direction, draw the line from the far pole — the point diametrically opposite the touch point — through the direction’s place on the sphere, and continue it until it meets the plane. That intersection is the mark. The near pole maps to the origin, the equator maps to a circle of radius 2, and the far pole maps nowhere at all, which is the one direction the surface has no image for.
The choice of the far pole as the centre is what makes the projection conformal, and the reason is a fact about a sphere that does not generalise: the two circles through a point on a sphere that make a given angle there are cut by the projecting cone in a way that is symmetric about the tangent plane, so the angle survives. Projecting from the centre of the sphere instead gives the gnomonic projection — the flat picture plane — which sends great circles to lines and destroys angles everywhere but one point. One change of the projection centre, and the surface swaps which of the two properties it keeps. That is the empty corner of the trade plot, seen from very close up.
The inverse is as short as the map and is used more often than it, because the rectification machinery elsewhere on this site goes backwards through a surface. Given a mark (u, v), set q = (u² + v²)/4; the direction is (u, v, 1 − q) divided by (1 + q). Pushing all six surfaces’ directions out and back returns them to within 3 × 10⁻¹⁶, which is checked rather than assumed — an inverse with a sign wrong produces a rectified picture that looks exactly like a rectified picture.
Circles to circles, measured by an instrument that was not told
Conformality is a statement about infinitesimally small figures. Stereographic satisfies a much stronger one that holds at any size: every circle on the sphere images as a circle or a straight line in the picture.
That includes the great circles, which are the images of straight world lines. So a straight line — a girder, a kerb, a horizon — appears in a stereographic picture as an arc of an exact circle. Not approximately, not near the centre: exactly, everywhere.
The measurement of this is the part worth dwelling on, because it uses an instrument built for a completely different essay.
The site fits general conics, Ax² + Bxy + Cy² + Dx + Ey + F = 0, by a smallest-eigenvector solve with a Jacobi eigensolver. That machinery exists because the image of a circle lying on the ground is an ellipse whose centre is not the image of the circle’s centre, and measuring the gap between those two points needed a fit that could not be fooled by the conditioning of a figure-sized canvas.
Handed the sampled image of a great circle, the same fit is asked one question: how far are the coefficients from A = C and B = 0, which is what a circle is? It has no idea which surface produced the points.
- Under stereographic it reports 4 × 10⁻¹⁰ of its own scale. A circle.
- Under a flat plane the fit degenerates and reports a straight line, to 3 × 10⁻¹⁵ pixels of collinearity. Which is correct: a flat picture plane is a gnomonic projection, and gnomonic sends every great circle to a line.
- Under a cylinder it reports 1.3. Neither a circle nor a line, and not close to either.
Three surfaces, one instrument, three answers, all right. An instrument that gave the same answer to all three would be measuring nothing, and this is the property that makes the stereographic result worth quoting rather than restating.
The little planet, and why it works
The stereographic surface is what a photographer means by a little planet: a 360° panorama re-projected so that the ground curls into a disc with the horizon as its rim, and the sky wraps round the outside.
It is a striking picture and its striking quality has a geometric cause. Because the surface is conformal, every recognisable object in it keeps its shape — a person standing on the little planet is a correctly proportioned person, a window is a correctly proportioned window. Because straight lines become circular arcs, the architecture is curved while the objects are not. That combination is unique to this surface: on a fisheye the objects distort too, and on a flat plane there are no arcs to curve.
What the little planet gives up is area. Stereographic’s area scale grows without bound toward the antipode of the projection point, so a full 360° picture puts an enormous amount of paper into whatever happens to be directly behind the photographer — usually the sky, occasionally a tripod. That is why the little planet is always shown with the ground at the centre and the zenith stretched round the edge: the surface has one direction it treats badly, and the choice of which direction that is is the composition.
The straightest of the curved ones
There is a small result in that plot that runs against expectation, and it is worth naming because it is the sort of thing a survey turns up and an argument does not.
Of the five curved surfaces, stereographic bends a straight line the least. The test line — a ground line eleven metres to either side of the view axis, six and a half metres in front — departs from its own chord by 3.5% of that chord on stereographic, against 4.5% on the equidistant fisheye, 5.1% on the equal-area fisheye, 5.7% on equirectangular and 5.9% on the cylinder.
That is not a coincidence and it follows from the circle property. A circular arc through two given endpoints, of the curvature stereographic produces at moderate field angles, is a shallower departure from the chord than the transcendental curves the other surfaces produce over the same span. The surface that keeps shape exactly also happens to be the one that disturbs straightness least, over the range where anybody would look.
It does not stay that way. Push the field of view toward a full turn and stereographic’s area behaviour makes it unusable long before its straightness does, while the equal-area fisheye is still perfectly well behaved. Every one of these comparisons is a comparison over a stated range, and the range is part of the claim.
Two exact zeros, and why exactness is the point
Across the whole six-surface comparison there are exactly two quantities that come out as zeros rather than as small numbers, and it is worth being clear about why that distinction carries weight.
Stereographic’s angle error is one: 1.7 × 10⁻⁹ degrees, which is the differencing noise and not a property of the surface. The equal-area fisheye’s area scale is the other: 1.000000 across the entire field, flat to the last digit the measurement can resolve. Everything else in the comparison is a curve that grows — the plane’s area scale, the cylinder’s anisotropy, the equirectangular surface’s everything.
A curve that grows can be argued about. Is 10% of anisotropy acceptable? Is 40%? Those are questions about a viewer and about a purpose, and they have no answers in geometry. A zero cannot be argued about. It says the surface has the property, not that it approximately has it over some range somebody chose, and it is the only kind of statement in this field that survives being pushed.
That is also why the two zeros are the two properties surfaces get named for. Nobody calls a surface “nearly equal-area”. They call it equal-area or they call it something else, because the property is the kind of thing a surface either has or does not. The intermediate surfaces — equidistant, cylindrical, equirectangular — are named after what they do to a coordinate rather than after what they preserve, and that is not an accident of naming: there is nothing they preserve to name them after.
The fleet habit that produced this comparison applies here too. A zero reported by a measurement that could only ever report a zero would be worth nothing, so the same machinery is pointed at the surfaces that must fail and required to say so. Loosen the definition of conformal far enough that the flat plane qualifies, and the impossibility check fires; give the angle test a right angle aligned with the cylinder’s own axes and it wrongly passes, which is the failure the rotated version exists to prevent. Both are in the site’s gate, and both reject.
Where else this surface turns up
Stereographic is not a specialist choice for photographers. It is one of the oldest projections in use, and the reasons for using it are the same in all of them.
Cartographers use it for the polar regions, because it is the conformal azimuthal projection — a coastline on a polar stereographic map crosses a meridian at the angle it really crosses it, which is what a navigator wants. Crystallographers use it for pole figures, because the angle between two crystal directions can be read off the plot with a ruler and a net. Complex analysts use it as the Riemann sphere, where it makes the point at infinity an ordinary point — which is exactly the move this site’s first field makes with the projective plane, arriving from a different direction.
The common thread is that all four uses want angles and none of them wants area. That is not four separate facts. It is one surface being chosen four times for the one thing it preserves, by people who each had a different reason not to care about the thing it destroys.
What it costs to use it as a picture
A surface with one exactly preserved property still has to be looked at, and the practical objections to stereographic are worth setting down beside the mathematics rather than after it.
The first is that its correct viewing arrangement is not a distance. Every viewing-distance number on this site — a 40° picture shown 160 mm wide is correct from 22 cm — is computed for a flat picture, where the geometry of being in the right place is simply a point on the normal through the centre. A stereographic picture printed flat is a picture on the wrong surface, in the same way that an unrolled cylinder is, and there is no single distance that repairs it. There is a viewing construction, and it involves a curved surface nobody is going to build.
The second is that its field of view has no natural stopping point and therefore has to be chosen. A flat picture announces its own limit: past about 120° it becomes unusable and past 180° it does not exist. A stereographic picture at 200° is perfectly well behaved, at 300° is still well behaved, and at 355° is a little planet with an enormous stretched sky. Nothing in the surface says where to stop, so the photographer decides, and the decision is invisible in the result — which is precisely the complaint this site makes about the classical constructions in the fifteenth century, arriving six hundred years later in a different tool.
The third is the one that keeps it out of general use: it is not what a lens does. Real wide lenses are built close to equidistant or equal-area, because those mappings are what a sequence of refracting surfaces produces naturally, and stereographic fisheyes exist but are unusual and expensive. So a stereographic picture is almost always a re-projection of something else, which means it carries whatever the original surface lost. Conformality restored in software does not recover a shape the sensor never recorded.
The limit worth stating
Everything above is about geometry, and the honest boundary is the one this site draws everywhere.
A conformal surface preserves the shape of infinitesimally small figures exactly, and the shape of finite ones only approximately. A face occupying a degree of the field is preserved to a part in ten thousand; a building occupying ninety degrees is not preserved at all, and calling stereographic “shape-preserving” without that qualifier is how the word gets misused. What is exactly preserved at every size is not shape but circularity, and those are different claims.
And the perceptual half is not addressed here at all. A little planet does not read as a correct picture to anybody, despite being conformal, because the visual system reads curved horizons as evidence about the world rather than as evidence about the projection. That is a fact about seeing, and it is not made less true by the measurement above — nor is the measurement made less true by it. They are answers to different questions, and the reason to keep them apart is that the geometric one is checkable and the perceptual one is not, and mixing them lends the second an authority it has not earned.