The condition does not always decide the map

A projection family is usually presented as a list. The cylindricals are Mercator, the plate carrée, Lambert’s equal-area and a few compromises; the conics are Lambert’s conformal, Albers’ equal-area and the equidistant one; the pseudocylindricals are Mollweide, sinusoidal, Eckert IV and thirty more with people’s names on them.

Two rungs of this ladder have already replaced part of that list with a parameter. The conic is the whole family shows the cylindrical and azimuthal families as the two ends of one cone constant; the azimuthal family is one function shows five azimuthal projections as five choices of one radial function, each picked out by a property.

This rung finishes the job and finds that the answer is not the same in every family. Write the family as a shape with an unknown function in it, state the property as an equation, and solve. Three families answer with one map. One answers with an infinity of them.

Three conditions, one unknown function #

The cylindrical ansatz is x = λ, y = Y(φ). Every scale factor follows: the scale along a parallel is k = sec φ, fixed by the shape and not by the choice of Y, and the scale along a meridian is h = Y′(φ), which is the choice. So:

• equidistant along meridians means h = 1, so Y′ = 1 and y = φ — the plate carrée;
• conformal means h = k, so Y′ = sec φ — Mercator;
• equal-area means hk = 1, so Y′ = cos φ — Lambert’s cylindrical equal-area.

Three lines, three projections, no formula transcribed. The integrations agree with the closed forms to 2 × 10⁻¹⁶, 1.5 × 10⁻¹² and 3.7 × 10⁻¹⁴ of the ordinate respectively, which is the arithmetic’s floor in each case.

The conic ansatz is x = ρ sin nλ, y = ρ₀ − ρ cos nλ with ρ = R(φ) and a cone constant n. Again one unknown function, again three first-order equations: R′ = −1, R′ = −nR sec φ, R′ = −cos φ/(nR).

Nothing in that table is quoted. Each map is an integration, and the properties are measured afterwards by the machinery that measures every other projection here — which is what makes the equal-area conic’s areal spread of 1.4 × 10⁻¹⁰ a result rather than a definition.

The one that is nonlinear, and what that costs #

The equal-area conic’s equation is the only one of the nine in which the unknown appears on the right-hand side, and it is the only one that can leave the map’s own domain. R = 0 is the cone’s apex, and an integration that reaches it keeps going: the arithmetic works perfectly well at negative radii, where the map becomes a reflection of itself and still looks like a map.

The first version of this measurement started every integration at R = 1. The equal-area solution reached the apex at 40° north and continued, and the areal spread it reported was 1.8 × 10⁻⁴ at four thousand steps and 2.58 at sixteen thousand — a quantity moving by four orders of magnitude with the step size, which is what a measurement taken on the far side of a singularity looks like and is not what a failing integration looks like. Each condition now starts where its own closed form starts, which is what a cone of that constant is actually the size of.

The fourth family has two unknown functions #

The pseudocylindrical ansatz is x = λ·C(φ), y = Y(φ): parallels straight and evenly divided, meridians curved, the pole drawn as a point or a line. It has two unknown functions, and the equal-area condition is still one equation:

$$C(\varphi)\,Y'(\varphi) = \cos\varphi$$

One equation in two unknowns does not determine anything. Choose Y freely, and C follows.

The sinusoidal comes from Y = φ; the cylindrical equal-area comes from Y = sin φ; the third comes from Y = φ/2 + sin(φ)/2, chosen for no reason at all and equal-area to eleven decimal places. Mollweide comes from an auxiliary angle satisfying 2θ + sin 2θ = π sin φ, and satisfies the same equation, and so does Eckert IV, and so do the thirty others.

That is the answer to a question the subject usually treats as a matter of taste. The cylindrical family has exactly one equal-area member because its condition is an equation in one unknown. The pseudocylindrical family has as many as anybody cares to name because its condition is an equation in two. Nobody has to invent a new principle to explain why one family is crowded and the other is not; the count of unknowns explains it.

A condition can also empty a family #

The third thing a condition can do to a family is leave it with nothing.

In the pseudocylindrical ansatz the ordinate does not depend on longitude: ∂y/∂λ = 0. Conformality in these coordinates is the Cauchy–Riemann pair — x_λ / cos φ = y_φ and y_λ / cos φ = −x_φ — and the second of those, with y_λ = 0, forces x_φ = 0. But x = λ·C(φ), so x_φ = λ C′(φ), and requiring that to vanish for every longitude forces C′ = 0.

C constant means the meridians are straight and evenly spaced: the map has left the pseudocylindrical family and become cylindrical. The first condition then reads Y′ = C sec φ, which is Mercator up to scale.

So a condition applied to a family can fix it, leave it free, or empty it, and which of the three happens is decided by counting: how many unknown functions the ansatz has, and how many independent equations the condition imposes.

What was computed, and how #

Nine integrations and one algebraic argument.

Each cylindrical and conic condition is a first-order ordinary differential equation, integrated by classical fourth-order Runge–Kutta at four thousand steps and tabulated. The pseudocylindrical solutions need no integration at all — C follows from Y by division — so they are exact functions.

The tabulated solutions are then handed to the site’s ordinary distortion machinery, which differentiates a projection numerically at a step of 10⁻⁴ radians. That created the one trap worth recording. The table’s spacing is 0.044° and the derivative’s step is 0.0057°, so a linearly interpolated table is a polygon at the scale the derivative works at: the solved conformal conic came back with 0.076° of angular deformation — not conformal by this site’s own tolerance, entirely because of the interpolation. The value being interpolated is the solution of a differential equation, and the equation is its derivative, so a Hermite cubic through two values and two known slopes costs nothing and is fourth order. The same measurement then returns 1.5 × 10⁻⁶ degrees.

That failure is the second time this collection has met the same shape — a lookup table read at a finer resolution than it was built at — and the fix is the same both times: interpolate with the derivative the generating rule already gives.

A parameter is not a free function #

The conic family looks at first like a counterexample to the counting. It has one unknown function and a parameter, so a single condition should leave a one-parameter family of solutions — and it does. What makes it different from the pseudocylindrical case is what that freedom is worth.

A parameter is a finite choice, and the members it produces are the same map at different settings: every conformal conic is conformal, and picking n decides where the two standard parallels fall and therefore where the distortion is least. A free function is an infinite choice, and the members it produces do not resemble one another — a sinusoidal and a cylindrical equal-area share a property and nothing else.

The practical difference is what a cartographer has to supply. Choosing a cone constant needs one number, which a region determines: the conic is the whole family shows the cylindrical and azimuthal cases arriving as the two limits of the same integration rather than as separate constructions. Choosing a parallel spacing needs a curve, and no region determines one.

The same method, in the family it was invented for #

The azimuthal family is the one this collection solved first, and it is the cleanest instance of the count: one unknown function of one variable, one condition, one map. The azimuthal family is one function integrates the equations and finds the gnomonic, stereographic, equidistant and Lambert azimuthal projections in the same way the cylindrical ones are found above, with the orthographic as the member that satisfies none of the three conditions and is defined by a construction instead.

Set the four families side by side and the pattern is complete. Azimuthal: one function, one map. Cylindrical: one function, one map. Conic: one function and a parameter, a one-parameter family of maps. Pseudocylindrical: two functions, an infinity of maps — and, for the conformal condition, none at all.

Four families, one method, four different kinds of answer. That is a statement about the ansatz rather than about the projections, and it is the reason this rung exists as an argument rather than as a table.

Where the model stops #

An ansatz is a choice, and choosing it is where the real work of the subject sits. Nothing here derives the four shapes: they are written down because they are the four classical families, and a family nobody has written down is not reachable by this method. The polyconic is the standing example — the projection that gave up being one thing has a different cone for every parallel and therefore no ansatz of this kind at all.

The conditions are equally conventional. Equidistance here means along the meridians, which is one of several things the word is used for; a map equidistant from a point or along a set of lines is a different condition and is what a projection written as a condition takes up.

And the counting argument is about local freedom rather than about what makes a good map. The pseudocylindrical family’s extra function is a genuine degree of freedom and it is also the reason that family needs an aesthetic argument to choose within it: the condition does not decide, so something else must, and the something else is what where a pseudocylindrical puts its error is about.

What the four members of the free family do at the pole is exactly that freedom made visible: a point, a line, and everything between. The condition is silent about which, and the choice is paid for in the distortion pattern rather than in the property.

The generalisation #

The rule is a count and it is not cartographic. A property imposed on a parameterised family determines a member when the number of independent conditions equals the number of unknown functions, leaves a family of members when it is smaller, and empties the family when it is larger.

The three outcomes all appear here in one subject, at one rung, with the same machinery measuring all of them — which is unusual, and is the reason to state the rule in this form rather than as three separate observations about three families.

The version worth carrying away is narrower. A crowded family and a sparse one are usually explained by history: this one attracted more attention, that one was solved early. Sometimes that is right. Here it is not: the pseudocylindrical family is crowded because its defining condition is one equation short of determining a map, and no amount of attention would have made the cylindrical family crowded.

What the ansatz costs #

There is one price in this method that is easy to miss, and it is worth stating because it bounds what the counting argument can claim.

The ansatz is not derived, and every conclusion is conditional on it. Writing the cylindrical family as x = λ, y = Y(φ) builds in that the parallels are straight, evenly divided and horizontal, which is three assumptions before any condition is imposed. The equal-area member that comes back is the equal-area member of that shape, and there are equal-area maps with straight parallels that this ansatz cannot express because it fixes the longitudinal spacing.

The counting therefore says something narrower than it appears to. It says how much freedom a condition leaves within a stated shape, not how many equal-area maps exist. What makes the narrower statement worth having is that the shapes are the ones the subject actually names, so the count explains the sizes of the families anybody has to choose between.

Who found it, and when #

Lambert’s 1772 memoir derives the conformal conic and the cylindrical equal-area from their conditions, which is the method of this essay applied by hand two and a half centuries ago; Tissot’s later work makes the differential apparatus general. The particular observation that the pseudocylindrical equal-area condition leaves a free function is implicit in every derivation of Mollweide, Eckert and the rest — each of which begins by choosing a parallel spacing and then deriving the meridian shape — and it is rarely stated as the reason the family is crowded.

That a conformal map with straight parallels must be Mercator is older still and is usually derived from the isometric coordinate rather than from the Cauchy–Riemann conditions in this ansatz. Both routes are two lines; the one here is the one that fits the counting argument.

What the count is a count of #

The counting is done inside an ansatz, and that qualification carries more weight than it first appears to, so it is worth saying exactly what the resulting numbers are and are not.

They are not counts of projections. The pseudocylindrical equal-area condition leaves one free function is a statement about maps of the assumed shape — parallels straight and horizontal, meridians equally spaced along each. Drop the shape and the equal-area condition on its own leaves a free function of two variables, which is the whole of the equal-area family and is an incomparably larger object.

They are counts of freedom given a commitment. The ansatz is the commitment, the condition is applied inside it, and the count says what remains. That makes each number a joint property of a shape and a property, and it is why a family can be crowded, sparse or empty depending on which shape it is asked about — the same condition emptying one family and leaving a function free in another.

Which explains a pattern in how the subject is written. Different treatments disagree about how many projections a family “contains”, and the disagreements are almost never about the mathematics: they are about where the ansatz was drawn. A treatment that allows unequally spaced meridians along a parallel and one that does not are counting different sets, and both counts are correct.

What would make a count absolute is a statement about the space of all maps, with no shape assumed, and the subject has exactly one of those — the equal-area case above, where the answer is a free function of two variables and is therefore too large to be useful as a taxonomy. That is the trade the ansatz is making: it throws away most of the space in order to leave a count small enough to name the members of.

It also says what a reader should ask of any such count they meet elsewhere. Not how many members does this family have, which has no answer, but what was assumed about the shape before the condition was applied — and if a treatment does not say, the count it prints cannot be compared with anybody else’s.

So the right reading of the numbers in this essay is as a measure of how much design freedom a named family leaves after its defining property is imposed — which is the question a designer actually has, and which is why the narrower statement is the useful one.

Where the ladder goes next #

A family with a free function invites the obvious next question: choose the free function to make some other quantity as small as possible. That is an optimisation over a function rather than over a parameter, and this site has one piece of machinery of the right kind — the least-squares solver that fits a conformal map to a condition on a region’s boundary, which appears in a map with no formula. What the two have in common is that neither picks a projection from a list.