The lines a surface leaves alone

The curved field’s foundational result is that only the plane keeps straight lines straight, and this site asserts it directly: the flat picture bends a probe line by nothing, and every curved surface bends the same line by a measurable amount, with the straightest of them still off by a stated percentage of the chord.

That is true and it is read as more than it says. A reader takes it to mean that a curved surface bends everything, and it does not. Every one of them leaves a family of lines exactly alone, and which family it is turns out to be the most useful single fact about the surface — more useful than the bend, which is a number about one line, and more useful than the area range, which is a number about one field of view.

The question, stated so it has an answer #

A straight line in the world is a great circle on the sphere of view, and a great circle is named by its axis — a direction on the sphere, up to sign. So the set of world lines is a two-dimensional sphere of axes, and asking which lines a surface draws straight is asking about a subset of that sphere.

Subsets have dimension. If the answer is all of them, the set is two-dimensional. If the answer is a curve, it is one-dimensional. If nothing, zero.

That is the whole question, and stating it that way is what makes it answerable rather than a matter of examples.

The first version of this measurement was worthless #

The obvious way to find the set is to sample it: lay a grid of axes over the sphere, image each one’s great circle, and keep the ones whose image is straight to a tolerance.

Run that on the cylinder over a 64 × 64 grid and it reports 120 straight axes out of 4,096. Run it on the three fisheyes and it reports zero, and on the equirectangular surface zero.

None of those numbers is a measurement of anything. A one-dimensional curve on a two-dimensional sphere has no area, so a grid of points misses it except where a grid point happens to land on it — and 120 is the grid’s spacing, not the surface’s behaviour. Reporting zero for a surface that has a whole family of straight lines is the same defect with the accident absent.

This is the third time this site has recorded a measurement of that shape: a quantity evaluated where it cannot see what it is looking for. The first was a cross-ratio evaluated over four consecutive divisions, the second an angle differenced along the surface’s own axes, and this is the third, and all three passed everything at the time they were written.

Looking for a curve along paths #

The repair is to stop sampling the set and start crossing it.

A one-parameter path through the sphere of axes crosses a codimension-one set transversally, in isolated points, and misses nothing. So: take a family of great-circle paths through the sphere of axes, scan the straightness along each, refine every local minimum by golden section, and keep the ones that reach zero.

A surface whose straightness is zero everywhere is caught first, by a coarse sweep, because for that one the root-finder would return an arbitrary point of a region and call it a curve.

The verdict is then: dimension two if the coarse sweep says everything, dimension one if a root is found on essentially every path, and zero if no path finds one.

The plane returns two, at 100% of sampled axes. Every other surface here returns one, on 96% to 100% of paths, with under 0.05% of the sphere straight by area. The one surface that returns zero is the cube map, and its reason is its own.

The parameterisation nearly cost the answer a second time #

There is a second failure hiding behind the first, and it was live for one commit.

The natural way to scan for a curve on a sphere is to fix an azimuth and sweep the elevation. That works for the cylinder, whose family is the circle of horizontal axes — every azimuth crosses it. It fails completely for the azimuthal surfaces, whose family is the circle of axes perpendicular to the view direction: in that parameterisation the family lives at two particular azimuths and nowhere else, so a sweep in elevation finds a continuum of roots at 6% of azimuths and nothing at the other 94%.

The measurement reported “0.5-dimensional” for the three fisheyes, which is not a dimension. It was reporting how well the coordinate grid happened to line up with the answer.

Scanning along a family of deliberately skew paths removes the dependence, because a path chosen without reference to the surface’s own axes crosses a codimension-one set whatever the set’s orientation. That is the same principle as rotating the right angle in the conformality test, and it is the same principle as taking the four points of a homography fit from the design’s corners: a measurement must not be evaluated in coordinates the thing being measured helped choose.

What the families are #

Naming them makes the abstraction concrete, and each is a family a photographer already knows.

On a cylindrical surface — the cylinder and the equirectangular unrolled sphere — the straight family is the verticals, plus the horizon. A vertical world line has constant azimuth, the map’s first coordinate is the azimuth, so the image has constant horizontal position and is a vertical straight segment. The horizon is the one crosswise member.

On an azimuthal surface — stereographic, equidistant, equal-area — the straight family is the lines whose images run through the picture’s centre. A great circle passing through the optical axis has all its directions in one plane through the axis, and every azimuthal map sends such a plane to a radius.

On the plane it is everything, which is Beltrami’s theorem stated as a dimension.

The signature, and what it cannot do #

Dimension alone separates the plane from everything else and stops. What separates the two remaining groups is whether the straight images are concurrent — whether they all pass through one point.

They do on every azimuthal surface, and the fitted meeting point comes back at a spread of 1e-12 picture units for stereographic and around 2e-6 for the two whose roots are found less precisely. They do not on either cylindrical surface: the verticals are parallel and the fitted intersections run off to 1e+7 and beyond, which is the numerical form of “no meeting point”.

So the pair (dimension, concurrent) is a signature, and over the six surfaces it takes exactly three values: two-dimensional; one-dimensional and concurrent; one-dimensional and parallel.

Three values over six surfaces is a coarse invariant, and this is the honest yield of the measurement. It cannot tell the cylinder from the equirectangular surface, and it cannot tell the three fisheyes apart.

The limit is a result #

That last sentence is where the essay’s interest is, because a measurement that failed to distinguish things which are different would be a bad measurement, and this one is distinguishing things that are the same in the respect being measured.

The site already found, by a completely independent route, that the cylinder and the equirectangular surface have identically the same anisotropy, $\sec\phi$, to $10^{-9}$. They differ in area and in reach, not in shape. Now they are found to have identically the same straight family too.

Two independent measurements agreeing that two surfaces have the same shape is worth more than either alone. The straight family is not failing to separate them; there is nothing there to separate.

The three fisheyes are a different case and the measurement is honest about it. They differ enormously — stereographic is conformal and equal-area has an exactly flat area scale, two exact zeros in the whole comparison — and they differ in the radial profile of the map, which is precisely the thing a family of radial lines cannot see. The straight family is blind to the one axis along which those three surfaces vary.

What the family is worth as a classification #

Three values over six surfaces sounds like a poor invariant, and it is worth asking what a better one would look like before accepting it.

The field’s other measurements each give six different numbers, so each of them separates all six surfaces. That looks stronger and it is weaker, because the numbers depend on the field of view sampled: the bend, the worst angle, the anisotropy and the area range all change when the fan changes, and two surfaces can swap places. A ranking that moves is a ranking of the measurement.

The straight family does not move. It is a topological fact about the map — how many dimensions of lines survive, and whether their images meet — and it is the same at every field of view, at every focal length, and at every scale the picture is drawn at. It puts the six surfaces into three boxes and the boxes never change.

That is the trade a coarse invariant makes, and it is the same trade the group-level classification makes one field away: a map’s fixed structure has three cases and a map’s matrix has eight numbers, and the three cases are the ones that survive re-coordinatising.

The cube map’s zero #

The one surface returning dimension zero deserves its own paragraph, because the answer is right and means something different from every other zero.

A cube map draws a line exactly straight if the line’s image stays inside one face, and not at all if it crosses a seam. The probe used here spans about a 50° arc, and a 50° arc crosses a seam from almost anywhere — 2.8% of sampled directions avoid one. So over the probe’s own length there is essentially no straight family, and the measurement reports zero correctly.

Shorten the probe and the answer changes: at a 9° arc, 94.4% of lines are straight, which is dimension two over that length. The cube map’s straight family depends on how long a line is drawn, which is a scale rather than a family, and it is the reason the cube map is not one surface but six.

What a photographer does with it #

The families are the practical content, and each one is a compositional rule that usually arrives as folklore.

On a cylindrical panorama, keep the verticals vertical. They are the only lines the surface draws straight, so a picture composed around uprights — columns, doorframes, trees, the corners of buildings — has every one of its strong lines exact and everything else bowed. That is why cylindrical panoramas of architecture work and cylindrical panoramas of a level horizon at an angle do not.

On a fisheye, compose radially. The straight lines are the ones running out from the centre, so a subject arranged along a radius is drawn straight and one arranged across the frame is drawn as an arc. A photographer who has learned to put the horizon through the middle of a fisheye frame has learned that the horizon is a member of the family when it passes through the centre and not otherwise.

On the Panini family, the family is the cylinder’s, because the construction is built on a cylinder. Every member keeps verticals straight, which is the whole reason the family exists and is the sense in which it is a family selected for buildings.

The measurement stated as a rule #

Two sentences, and they replace the reading the field’s headline result usually gets.

Only the plane draws every line straight, which is Beltrami’s theorem and is what the site already asserted.

And every curved surface draws a one-parameter family straight, which nobody says, and which is the family a picture on that surface should be composed around.

The second is not a weakening of the first. A one-parameter family in a two-parameter set is a set of measure zero — a photographer choosing a line at random gets a bent one with probability one — and it is also every line of one useful kind. Both facts are true and only the first ever gets stated.

What would separate the three fisheyes #

Since the essay has stated a limit, it should say what removes it, and the answer is already in the field.

The three azimuthal surfaces differ in their radial profile: the picture radius as a function of the angle off axis. It is $2\tan(\theta/2)$ for stereographic, $\theta$ for equidistant, $2\sin(\theta/2)$ for equal-area. Every difference between them is in that one function, and every straight line in the family runs radially, so the family is exactly the set of lines along which the profile cannot be seen.

The measurement that does see it is the one the field already has — the area scale, or equivalently the anisotropy, along a radius — and it separates all three by a wide margin: equal-area’s area scale is exactly flat, stereographic’s runs away fastest, equidistant’s sits between them.

So the two measurements are complementary in a precise sense rather than a vague one. The straight family sees everything about the map except the radial profile; the distortion battery sees the radial profile and cannot see which lines are straight. Neither is a substitute for the other and together they determine the surface among these six.

The short version #

A world line is a great circle and a great circle is named by an axis, so the set of lines a surface draws straight is a subset of a sphere and has a dimension.

The plane’s is two-dimensional: every line. Every curved surface here has a one-dimensional family — a curve on the sphere of axes, with no area, which is why a grid of samples finds none of it and the first version of this measurement reported zero for four surfaces out of six.

The pair (dimension, concurrent) takes three values over six surfaces. It separates the plane from the rest and the azimuthal group from the cylindrical one, and it cannot separate the cylinder from the equirectangular surface — which the site had already found, independently, to have identically the same anisotropy.

A curved picture surface does not bend everything. It bends everything except one family, and the family is what a picture on it should be built from.