Every equal-area map is every other one

A projection written as a condition is a sentence rather than a formula, and the condition ladder has spent five rungs asking what different sentences produce. Two distances exact gives a construction of intersecting circles. Three gives an over-determined problem with a measurable spread. A property demanded over a whole region gives a least-squares solve with no closed form at all.

Underneath all of them sits an assumption that has never been examined: that a condition decides a map. It does not always, and how far it falls short is measurable.

The two classical properties are the natural place to ask, because they are the two conditions the whole subject is organised around and because they are not the same size as each other.

One equation on two functions #

A projection is a pair of functions of two variables: x(λ, φ) and y(λ, φ). Written as conditions, the two classical properties are not the same size.

Equal-area is one equation. The areal scale factor is the Jacobian determinant divided by the area element on the body, and demanding it be 1 is a single scalar constraint at each point:

x_λ y_φ − x_φ y_λ = cos φ

Conformal is two. Angles are preserved when the Jacobian is a scalar multiple of a rotation, which is the Cauchy–Riemann pair:

x_λ = y_φ cos φ and x_φ cos φ = −y_λ

Two functions constrained by one equation leave a function’s worth of freedom; two functions constrained by two equations leave much less. That is the counting, and everything measurable here follows from it — with one correction that the naive version of it gets wrong and that the rest of this essay depends on.

The freedom in the first case has a name and a shape: it is the group of area-preserving maps of the plane. Apply any of them after an equal-area projection and the composite is still equal-area, because the areal factor of a composition is the product of the two areal factors and one of them is 1 everywhere.

Both families are infinite, and the difference is how many arguments #

The counting above invites the conclusion that the conformal condition leaves a finite family, and it does not. The conformal maps of a region are the holomorphic functions of one complex coordinate, and there are as many of those as there are functions — so both conditions leave infinite-dimensional freedom, and the counting has to say something sharper than one equation against two.

What it says is how many arguments the free function has.

The equal-area condition is one equation on two functions of two variables, so what is left over is an arbitrary function of two variables — the shear’s amplitude may be any function of the row, and the row may itself be any of two coordinates’ worth of ground.

The conformal condition is the Cauchy–Riemann pair, and its solutions are rigid in a way the equal-area condition’s are not: a holomorphic function is determined by its values on the boundary, so the free choice is an arbitrary function of one variable rather than two.

Both families are infinite and one of them is a whole dimension smaller. That is the precise version of the intuition that conformality is the more restrictive property, and it is the reason Chebyshev’s criterion exists for conformal maps and has no equal-area counterpart: the criterion is a boundary condition, and a boundary condition can only pin down a family whose members are determined by their boundaries.

It also explains why the orbit table’s two halves look symmetric and are not quite. The plane map between two equal-area projections is an arbitrary area-preserving map — two coordinates’ worth of freedom, and it can be a shear. The plane map between two conformal projections is a holomorphic map, which is one coordinate’s worth, and it cannot shear anything: it is locally a rotation and a scaling at every point, which is exactly why its angular deformation comes back at 2 × 10⁻⁹ degrees while its areal factor runs to sixty-four thousand.

So the freedoms are not the same size after all, and the direction of the difference is the one the intuition had — it is only the amount that the naive counting overstated. Conformality removes a dimension’s worth of arguments and leaves a function of one variable; equal-area removes nothing of the kind and leaves a function of two.

Two deformations no library contains #

Two are used here, and both are deliberately absurd, because the point is that they cannot be dismissed as special cases.

The shear is (x, y) ↦ (x + a sin(f y), y). Its Jacobian is [[1, af cos(fy)], [0, 1]], whose determinant is 1 for every amplitude and every frequency. Nothing about it is small.

The swirl is a rotation about the origin through an angle that depends on the radius. A rotation preserves area whatever its angle, and making the angle a function of a quantity the rotation does not change keeps the determinant at 1.

What each condition actually pins down #

That is the whole finding in one picture. A quantity that does not move at all across a sixteenfold change in the deformation is a quantity the deformation has no purchase on; a quantity that runs from 88° to 154° over the same range is one the condition never constrained.

So “equal-area” is a claim about exactly one number and says nothing whatever about any other. Not little: nothing. Any shape, any angle, any scale pattern is compatible with it, and can be reached from any equal-area projection without leaving the property behind.

The library is one orbit #

The consequence for the actual named projections is stronger than it sounds. Since the plane map between two equal-area projections is B ∘ A⁻¹, and since both preserve area, that plane map preserves area too. So every equal-area projection in the library is every other one, composed with a member of the group.

The symmetry in that table is the reason to compute both halves.

• Between two equal-area projections, the plane map preserves area exactly and distorts shape by up to 168° — very nearly the maximum a plane map can.
• Between two conformal projections, the plane map preserves shape exactly and changes area by a factor of 64,000.

Neither condition is stronger than the other. Each pins down exactly one of the two invariants of the plane map between two of its members, and leaves the other one entirely free. A reprojection is a map of the plane established that the areal factor is the one quantity that divides exactly between two projections; this is the same measurement read as a statement about what a condition is worth.

What decides which equal-area map, then #

Everything except the condition.

Mollweide, sinusoidal, Eckert IV, Hammer, Lambert cylindrical and Gall–Peters are all equal-area, and all six differ from one another by members of the same group. What separates them is a second criterion, which is never written in the same sentence as the property: a pole line of a chosen length, a boundary that is an ellipse, parallels that are straight, a particular compromise on shape distortion at mid-latitudes.

The same freedom shows up in miniature inside a single family. A pseudocylindrical projection draws the pole as a point or as a line, and the length of that line is free: the equal-area condition then decides the parallel spacing from it rather than the other way round. Choosing the pole line is choosing the map, and nothing is being traded away when it is chosen.

The condition does not always decide the map found the same thing inside a shape ansatz — that the pseudocylindrical family’s equal-area condition leaves a whole function free, which is why that family has forty members and the others have three. This rung says the ansatz was doing most of the work. Drop it and the freedom is not a function inside a family; it is the whole group, and every family is in the same orbit as every other.

The practical form of the statement is worth having plainly. A caption saying “equal-area” has told a reader one thing, and the thing it has told them is not that the map is any good. The projection that shows true size is about the same word being used as a recommendation; this is what is underneath that.

The number that ought to be in the caption #

If “equal-area” does not distinguish between them, something has to. The candidates are all criteria over a region, and the site has three of them: Airy’s, Kavrayskiy’s, and the maximum angular deformation. Each returns a different ranking of the same six maps, which is what a compromise projection is for — and none of them is implied by the property in the name.

That is the shape of the whole subject in one line. A projection’s name states the constraint it satisfies; a projection’s quality is entirely in the objective it minimises subject to that constraint, and the name does not carry the objective. No essay here may say a projection is best without naming the purpose, and this rung is the formal reason why: the constraint leaves a group’s worth of maps, and only the purpose picks one out of it.

Why conformality feels more decisive, and is not #

Cartographic writing treats conformality as the more restrictive property, and the intuition is not baseless: two conformal projections do look more like each other than two equal-area ones. But that is a fact about which conformal maps anybody publishes, not about the condition.

The conformal maps of the sphere are the analytic functions of one conformal coordinate — a conformal map onto a face uses exactly that — and the analytic functions are an infinite-dimensional family too. Mercator is the logarithm of the stereographic coordinate; the Peirce quincuncial is an elliptic integral of it; the Lagrange projection is a power of it. Any holomorphic function whatever gives another conformal projection, and most of them are unusable.

The measurement in the orbit table says so directly: the plane map between two conformal projections changes area by a factor of sixty-four thousand, which is not a small residual freedom. It is the same amount of freedom the shear had, expressed in the other invariant.

What makes conformality feel tighter is that its freedom acts on the page in a way the eye reads as a change of view — analytic maps look locally like rotations and scalings — while the area-preserving group’s freedom includes shears, which the eye reads as damage. The two groups are the same size. Only one of them is polite.

What was computed, and how #

The deformed projections are built by wrapping a library projection so that its forward map is composed with the plane map, leaving everything else — the metric, the visibility rule, the limit — untouched, so that the distortion machinery measures the composite with exactly the code it measures anything else with.

The areal and angular measurements are the site’s standard ones, sampled over a grid to 60° of latitude rather than to the pole. That restriction is stated because it matters: Mollweide’s own angular deformation runs to 129° at its pole, where the projection is degenerate for reasons that have nothing to do with any deformation, and a worst case taken there measures the pole every time.

The orbit table composes each pair through compositeDistortion, which builds the plane map’s Jacobian as J_B J_A⁻¹ with both taken against the same longitude and latitude, so the body’s metric cancels exactly and the result is a property of the two projections alone. Samples are taken at cell centres, never at ±180°, because every cylindrical projection is cut there and a Jacobian taken across the cut measures the cut.

Two tolerances are used, and the difference between them is a measurement rather than a convenience. The shear’s areal departure is 1.3 × 10⁻¹¹ at every amplitude from 0.2 to 3.2 — it does not move. The swirl’s grows with amplitude, from 2.6 × 10⁻¹⁰ to 6.6 × 10⁻⁸. A quantity that is analytically constant and numerically rising with the severity of the map is the numerical Jacobian losing digits, not the map losing the property, so the swirl is held to a looser tolerance and the reason is written where the tolerance is.

The assertions refuse as well as confirm: the deformation must leave the areal factor at its floor and must at least treble the angular deformation at a stated point; and the same deformation applied to Mercator must destroy conformality, which excludes “a plane deformation does not matter” as an explanation of the first half.

What the rung changes about reading a caption #

The practical consequence is small to state and hard to unlearn.

A property in a projection’s description is a constraint and not a recommendation. Equal-area says the areal factor is 1; it says nothing about shape, about angles, about how the distortion is distributed, or about whether the map is any good for anything. Conformal says the angular deformation is 0 and says nothing about area. Each is exactly one number pinned out of two, and the other number is free over a group large enough to contain maps nobody would print.

This collection’s own rule — that no essay may say a projection is best without naming the purpose — was adopted as an editorial discipline. It turns out to be forced: the constraint leaves an orbit, and only the purpose picks a member of it. A caption that names the property and stops has told a reader which orbit the map is in, which is a fact about it and is not the fact anybody wanted.

Where the model stops #

The group of area-preserving plane maps is exhibited here by two of its members, not characterised. Its Lie algebra is the divergence-free vector fields on the plane, which is infinite-dimensional, and nothing here proves that the equal-area projections form a single orbit under it — only that any two of the ones in this library are related by a member, which is a construction rather than a proof.

The maps used are all globally defined and smooth. An area-preserving map that folds, or that is defined only on part of the plane, would need the foldedness machinery this ladder built at its third rung, and would raise the question of whether the composite is a projection at all.

And the counting argument — one equation against two — is a statement about the local structure of the constraint, not a theorem about the solution spaces. It is the right way to see why the two conditions behave differently and it is not a substitute for the measurement, which is why the measurement is here.

It is worth adding that the freedom is not a defect in the definitions. A condition that determined a unique map would be a condition with no design left in it, and the whole practice of choosing a projection exists because the conditions leave room. What this rung measures is how much room — and the answer, for both of the classical properties, is all of it in one direction and none in the other.

Who found it, and when #

That the equal-area condition is one equation and the conformal condition is two is visible in Tissot’s 1881 treatment and in every derivation since. That the plane maps preserving area form a group is elementary. Putting the two together to say therefore the equal-area library is one orbit seems not to be said anywhere, presumably because nobody has any use for a sheared Mollweide.

The nearest published relative is the observation, standard in the pseudocylindrical literature since Tobler’s work in the 1960s, that the equal-area condition on a pseudocylindrical leaves the parallel spacing free — one function, chosen by taste. That is this result restricted to one family, and it produced the forty-odd members of that family by people making the choice differently.

Where the ladder goes next #

Six rungs have written projections as conditions and measured what the conditions produce. The next question the ladder owes is the one it has been avoiding since its second rung: when a condition is over-determined and cannot be satisfied, what is the right thing to minimise — and whether the answer depends on the condition or only on the region.