The developable surface was never necessary

Every account of map projections opens the same way. Wrap a cylinder round the globe, or set a cone on it, or lay a plane against it; shine a light from the centre; trace what falls on the surface; unroll it. This ladder’s own base essay opens that way, and it is the first thing anyone learns about the subject.

It is a story about construction, and the question this rung asks is whether it is true of the projections in use. Take a named projection and ask: is there a point from which a light could shine to produce it? For most of them there is not, and the answer can be settled by fitting one number.

One formula, both families #

Put the projection centre a distance hR from the axis in the meridian plane and project onto a tangent cylinder. The ray from (−hR, 0) through (R cos φ, R sin φ) meets the cylinder where the radial coordinate is R, and it meets it at height

y = R sin φ · (1 + h) / (cos φ + h)

Now put the centre a distance hR from the centre along the axis through the point of tangency, project onto a tangent plane, and the radius at angular distance c is the same expression with c in place of φ.

One formula, both families. That is a fact about perspective rather than about cylinders: the surface only decides whether the answer is read as a height or as a radius, and the family of centres is the same half-line either way. It is the same collapsing this ladder found when it discovered that the azimuthal family is one function and that the conic is the whole family, arriving one level further down.

Three values of h are the projections everybody knows. At h = 0 the centre is at the sphere’s own centre and the formula is tan: the central cylindrical projection, and the gnomonic. At h = 1 the centre is at the far side and the formula is 2 tan(·/2): the spacing of Gall’s stereographic and, on a plane, the stereographic projection itself. At h = ∞ the rays are parallel and the formula is sin: the Lambert cylindrical equal-area and the orthographic.

Fitting one number to eleven projections #

The fit is one-dimensional. The scale is a nuisance parameter and can be solved exactly at each h by ordinary least squares, so what is searched is the single number, on u = h/(1 + h) so that the whole half-line including infinity fits in a unit interval and the three landmarks land at 0, ½ and 1 exactly.

The six that fit are the gnomonic at h = 0 exactly, the stereographic at h = 1 exactly, and the orthographic together with the Lambert cylindrical, Gall–Peters and Behrmann projections at h = ∞. That last group is a single fact wearing three names: a cylindrical equal-area projection at any standard parallel has parallel spacing proportional to sin φ, and the standard parallel is a scale factor, which the fit removes.

The five that do not fit are Mercator, Miller, the plate carrée, the Lambert azimuthal equal-area and the azimuthal equidistant. Their best residuals run from 1.3 × 10⁻³ to 7.1 × 10⁻³ relative — small enough that a picture of the fit and the projection would look identical, and four orders of magnitude above the noise floor the exact ones sit at.

The coincidence that is not one #

Two of the five failures return the same answer: the plate carrée and the azimuthal equidistant both fit h = 1.7633, with the same relative residual of 1.30 × 10⁻³.

That is not a coincidence and it is the sharpest illustration of what the ansatz is doing. The plate carrée’s parallel spacing is y = φ. The azimuthal equidistant’s radial function is ρ = c. They are the same function of their own argument, so they pose the fitting problem the same problem, and the fit — which knows nothing about cylinders or planes — returns the same number.

The two projections could hardly look less alike. One is a rectangle and the other is a disc; one is the default nobody chooses and the other is the standard for a range chart. What they share is the property the ansatz can see: the spacing rule, which is all a perspective construction is, and which is indifferent to the surface it is read off — in the same way that the invariants of a projection are indifferent to its graticule.

The azimuthal family’s radial functions are one function of one parameter, and three of the five members are perspective while two are not. The two that are not are exactly the two whose radial functions were chosen for a property — equidistance and equal area — rather than traced from a light source.

Why the ones that fail, fail #

The pattern in the failures is not random and it is the essay’s point.

Mercator is defined by a differential equation: the parallel spacing is whatever makes the map conformal, which integrates to ln tan(π/4 + φ/2). No rational function of sin and cos equals a logarithm, so no centre can produce it. Its best fit, h = 0.341, is a compromise that is 0.7 per cent out.

The Lambert azimuthal is defined by an area condition, giving ρ = 2 sin(c/2). The azimuthal equidistant is defined by a distance condition, giving ρ = c. The plate carrée is defined by nothing at all — it is the identity on the coordinates.

Miller’s is a deliberate compromise: Mercator’s formula applied to four-fifths of the latitude and stretched back, chosen so the map looks acceptable at the poles, which is a compromise projection doing what compromise projections do. It fits h = 0.775 at 0.1 per cent, better than Mercator does, which is a comment on how little a good perspective fit means.

So the six projections that are perspective are the ones that were constructed rather than specified, and the five that are not are the ones that were specified by a property: conformality, equal area, equidistance, or acceptability. That is the transition this ladder has been describing from its other end — the projections that gave up being one thing, and the condition that does not always decide the map — seen as a fact about history rather than about mathematics.

What the surface does buy #

Dismissing the construction would be an over-correction, so here is the part of it that survives the audit.

The aspect. Rotating the surface before projecting is the whole content of an oblique or transverse map, and it is intelligible as a rotation of a physical object in a way that “compose the projection with a rotation of the sphere” is not. The formula is the second one; the picture that makes it obvious is the first.

The standard parallel. A secant cylinder cuts the sphere in two circles and those circles are drawn at true scale — which is exactly what a standard parallel buys, and which is a fact about the construction that happens to survive into the projections that were never constructed. The secant version of a non-perspective projection is defined by rescaling, and it is the same map.

The cut. A cylinder has to be slit before it can be laid flat, a cone has to be slit, and a plane does not — which is why cylindrical and conic maps have an edge somewhere and azimuthal ones have only a boundary. That is a topological statement and it is true regardless of whether any light was involved.

What the surface does not buy is membership. A projection is not cylindrical because it was made on a cylinder; it is called cylindrical because its meridians are equally spaced parallel straight lines and its parallels are perpendicular to them. That is a description of the output, and the classification everybody uses has been a description of the output since Lambert.

What was computed, and how #

The spacing function of each projection is read out of the projection’s own forward map, not from a formula written down beside it. For a normal cylindrical projection that is the northing along the central meridian; for an azimuthal one it is the distance from the centre, taken along the prime meridian — and taking it from the pole instead, which is what the polar aspect everybody draws suggests, measures a radius from the wrong origin and returns a plausible h for every projection in the library. That mistake was made here first, and it produced an entirely convincing table in which the gnomonic was orthographic.

The search is a grid of 2,001 values of u followed by a golden-section refinement, and the residual reported is relative to the projection’s own range, because an absolute residual in map units is not comparable between a projection whose pole is at 2 and one whose pole is at infinity.

The check that the ansatz is the family it claims to be is separate from the audit: at h = 0, 1 and ∞ the formula is required to reproduce the gnomonic, the stereographic and the orthographic to a relative departure below 10⁻⁹, and it does, at every sampled distance. Without that, “six of eleven fit” would be a statement about a curve-fitting exercise.

There is a second collapsing already in hand, and the two are independent. The cone constant runs continuously from the azimuthal case to the cylindrical one, so the three surfaces are one family in a different sense — a limit rather than a construction — and neither collapse needs a developable surface to exist.

Drawn side by side, the four azimuthal projections give the eye nothing to sort them with. Three are what a light source at some distance actually produces and the fourth is what a ruler produces, and no distance reproduces it — the sorting is a property of the spacing rule rather than of the appearance, which is why it took a fit rather than a look.

Where the model stops #

Normal aspect only. The fit compares spacing functions, which presupposes a projection with a spacing function — a normal cylindrical or a polar azimuthal one, up to the aspect rotation that this ladder showed is a free choice. A pseudocylindrical or a polyconic projection has no single spacing function and is not in the table.

Perspective from a point on one axis. The construction searched is the classical one: a centre on the axis of symmetry, a tangent surface. Widening it — a secant surface, an oblique centre, a non-spherical generating body — adds parameters and would fit more projections, and would also stop being the story the textbooks tell. The narrow ansatz is the one being tested.

The sorting is of this library. Eleven projections is a small sample, chosen because they are the ones this collection draws. A wider catalogue contains other genuinely perspective maps — the vertical near-side perspective at finite altitude is the family’s most-used member and is exactly the h between 1 and ∞ that a satellite sits at — and it contains a great many more specified ones. The proportion would move; the sorting rule would not.

And “fits” means to arithmetic noise. The six exact fits come back below 10⁻¹⁶ relative, which is the double-precision floor; the five failures are at 10⁻³, three orders above where a plausible-looking fit would sit. There is no ambiguous case in this library, which is convenient and is not guaranteed for a larger one.

The generalisation #

A construction that explains a family is not the same as a definition of its members, and a field can keep teaching the first long after the second has replaced it. The cylinder is a genuinely good explanation: it makes the aspect intelligible, it makes the standard parallel intelligible, and it produces four of the projections in ordinary use.

What it is not is a classification, and it is used as one. A projection is called cylindrical because its graticule is a rectangle, not because anything was wrapped round anything, and the two criteria pick out different sets — Mercator is in the first and not the second.

The pattern outside cartography is the one where a founding metaphor becomes a taxonomy. Chemistry’s orbital pictures are a construction that explains four elements’ worth of behaviour and is taught as a classification of all of them. The point is not that such metaphors should be dropped; it is that the moment somebody asks which members actually satisfy the construction, the answer is usually a minority, and that answer is worth knowing before the metaphor is used to decide something.

Who found it, and when #

The perspective projections are the oldest ones. The gnomonic is attributed to Thales in the sixth century BC and is the projection of the earliest sundials; the stereographic is Hipparchus’s, around 150 BC, and is the projection of the astrolabe; the orthographic was known to the Greeks as the analemma. All three predate any theory of what a projection is, because all three are what a light source actually does.

Everything else came later and came from a specification. Mercator’s 1569 map was constructed graphically to make rhumb lines straight, and Edward Wright gave the integral in 1599 — the first projection in history defined by a condition rather than by a construction. Lambert’s 1772 memoir introduced four projections in one paper, each derived from a property, and it is the moment the subject changes character.

The observation that the perspective family is one two-parameter formula is standard and appears in Snyder’s Map Projections: A Working Manual as the “general vertical perspective” and its cylindrical counterpart. What does not appear anywhere this collection has looked is the audit: taking the formula and asking of each named projection whether it is in the family, and finding that the answer sorts the library exactly by whether it was traced or specified.

What the sorting says about how the subject is taught #

The audit sorts the library into the projections a surface actually explains and the ones it does not, and the sorting is almost exactly the sorting into traced and specified. That is worth following through, because the developable-surface story is the first thing anybody is told about this subject and it is a description of the smaller half.

For the traced projections the story is literally true. The gnomonic, the stereographic and the orthographic are what a light source at three different positions does to a transparent globe, and the surface is not a metaphor — it is the paper, and the projection is the shadow. Those three are also the oldest, which is why the story exists.

For everything designed since 1569 it is a picture with nothing behind it. Mercator’s projection is the solution of a condition, and the cylinder in the diagram is a mnemonic for the shape of the result rather than an account of how it was constructed. There is no light source that produces it, no position for one, and no surface that can be wrapped and unrolled to give it. The same is true of every equal-area projection, every conformal one but the stereographic, and every compromise.

So the story teaches the exception as the rule, and the cost is a specific confusion rather than a vague one. A reader who believes a projection is a wrapping believes the family determines the map — that there is the cylindrical projection — and the subject’s whole difficulty is that a family is a shape and a projection is a shape plus a condition. That is the confusion this ladder’s other rungs keep having to undo: the conic being one parameter rather than three families, the pseudocylindricals being a free function rather than a list of names.

A replacement is available and is barely longer. A projection is a pair of formulas taking a place on the globe to a point on a page, chosen to satisfy a stated condition. Some of the oldest ones happen to be what a lamp does, and the surfaces in the diagrams are pictures of those — useful for seeing which way a family’s distortion runs, and not an account of where any of the maps came from.

The diagram is worth keeping under that reading. It says correctly that a cylindrical projection’s distortion is organised about a great circle and a conic’s about a small one, which is the fact this collection uses to count how many numbers an aspect has. It is a picture of the symmetry, and the symmetry is real even where the wrapping never happened.

Where the ladder goes next #

This rung asks what a family’s founding construction is actually true of. The datum ladder next door has a comparable habit to examine: it has printed a transformation’s seven parameters as exact numbers for nine essays, and every published set of them arrives with standard errors that are as much a part of the result as the parameters.