The azimuthal family is one function

The azimuthal projections come with a story each. The gnomonic is the view from the centre of the Earth; the stereographic is the view from the point opposite where the paper touches; the orthographic is the view from infinitely far away. Two more are defined by conditions rather than by viewpoints — one preserves distances from the centre, the other preserves areas — and the five are presented as a list.

They are not a list. They are one projection with a parameter, and the parameter is a function.

The reduction, in one paragraph #

An azimuthal projection is defined by two properties: the image of a point depends only on its angular distance from the centre and its azimuth, and azimuths are preserved exactly. That fixes everything except one number per distance — how far out to draw it.

Call that function $f(\rho)$. Then the whole projection is

$$x = f(\rho)\sin\alpha, \qquad y = f(\rho)\cos\alpha$$

and the five names are five choices:

Every one of them has $f(\rho)\to\rho$ as $\rho\to 0$, which is the statement that all five are true to scale at the centre. That is the only thing they agree about.

The list is worth reading as five answers to one question rather than as five formulae. Two of them are tangents of the distance and of half the distance; two are sines of the same pair; one is the distance itself. Halving the argument is what turns a projection that diverges at 90° into one that reaches the antipode, and swapping a tangent for a sine is what turns a projection that grows without bound into one that stops — so the four named perspective projections are the four combinations of two binary choices, and the fifth is the case where neither is made.

That is a piece of structure the viewpoint story cannot see at all, because the viewpoints are at four unrelated places and the arithmetic is at two.

The two scales, without differentiating a map #

Reducing the projection to one variable reduces its distortion to one variable too, and the two principal scales become derivatives of $f$ rather than of a two-dimensional map.

Along the radius, a step $d\rho$ on the ground becomes $f'(\rho)\,d\rho$ on the page, so the radial scale is

$$h = f'(\rho).$$

Around the circle, a circle of angular radius $\rho$ on the sphere has circumference $2\pi\sin\rho$ and is drawn with circumference $2\pi f(\rho)$, so the transverse scale is

$$k = \frac{f(\rho)}{\sin\rho}.$$

Nothing has been differentiated in two dimensions and no Jacobian has been formed. These are the two numbers the whole family’s behaviour is made of:

The patterns are the definitions read off. The stereographic’s two columns are equal at every distance; Lambert’s product is exactly one at every distance; the equidistant’s $h$ is exactly one; the orthographic’s $k$ is exactly one, which is the projection’s own definition seen from an unfamiliar angle — it preserves distances around the centre rather than away from it.

The check that makes it a reduction rather than a story #

The one-dimensional scales above must be the two-dimensional ones. This site’s ordinary machinery differentiates the forward map numerically against the sphere’s metric and extracts the singular values, which is a completely different calculation, and the two are required to agree.

They agree to $5\times10^{-12}$ across all five members at four distances each, which is the noise floor of the Richardson-extrapolated differences.

That is what turns the reduction from an appealing account into a fact. Without it, the essay would be describing a parameterisation that happened to reproduce five familiar formulae; with it, the parameterisation is the projections.

Each named property is a differential equation #

Once the scales are $f'$ and $f/\sin\rho$, the conditions write themselves.

Conformal means the two scales are equal:

$$f' = \frac{f}{\sin\rho}$$

Equal-area means their product is one:

$$f\,f' = \sin\rho$$

Equidistant means the radial scale is one: $f' = 1$, which is not a differential equation so much as an answer.

Each of the first two is a first-order equation with the same initial condition — $f(\rho)\to\rho$ near the centre, which is true scale there — and neither is told what it is supposed to reach. Integrating them by Runge–Kutta from $\rho_0 = 10^{-4}$ gives:

• the conformal equation, at every sampled distance out to 2 radians, within $8.5\times10^{-10}$ relative of $2\tan(\rho/2)$;
• the equal-area equation within $4.2\times10^{-10}$ of $2\sin(\rho/2)$.

So the stereographic projection is not “the view from the antipode that happens to be conformal”. It is the solution of the conformality condition among azimuthal projections, and there is exactly one. The same for Lambert’s and equal area. The viewpoints are a coincidence of the geometry, and the conditions are the definitions.

The member with no condition at all #

The gnomonic is the one member of the five that solves no condition, and its radial function explains both the property it does have and the price.

$f = \tan\rho$ is the only function for which a great circle on the sphere maps to a straight line on the page — the projection is a central perspective, and a great circle lies in a plane through the centre, so its image is the intersection of that plane with the page. Straightness is not a scale condition and does not appear anywhere in $h$ or $k$; it is a statement about which curves map to which, and the family’s reduction has nothing to say about it.

What the reduction does say is the price. $f' = \sec^2\rho$ grows without bound, so at 60° the radial scale is 4 and the transverse 2 — the indicatrix is twice as long as it is wide and eight times its proper area — and beyond 90° the function is negative and the projection has stopped existing.

The refusal, which is the equidistant member #

A construction that recovered the right answers from the right equations could still be a routine that returns whatever it is asked for. The check that stops that is a member which satisfies neither condition, measured against both.

The azimuthal equidistant projection has $h = 1$ and $k = \rho/\sin\rho$, so at 60° the two scales differ by 21 per cent and the areal factor is 1.209. Both residuals are of order one — the machinery reports 0.287 as the worst — and no tolerance anywhere on this site would accept them.

That member is not a failure. It is the projection whose own condition is the one being satisfied, and the family is large enough that there is one for every condition somebody has cared about. What the refusal establishes is that the two integrations landed where they did because of the equations rather than because of the integrator.

It is also the member with the most familiar use, which is worth noticing. Distances from one point being true is exactly what a range chart needs — an aircraft’s reach, a transmitter’s coverage, the polar chart on the flag of the United Nations — and the property is genuinely useless for anything else: distances between two points neither of which is the centre are not preserved at all, and at 80° from the centre the transverse scale is 1.418, so a ring drawn at that distance is 42 per cent too long. The shortest route is not straight makes the same distinction for the great circle; this family makes it for the ruler.

What the reduction says about the family as a taxonomy #

The conic essay on this site makes a structural claim: the cylindrical and azimuthal families are the two limits of the conic family, so the traditional three-way taxonomy is one continuous parameter read coarsely. The conic is the whole family measures both limits and finds them exact.

This is the same argument one level down, and it goes the other way. The azimuthal family is not a point in the conic parameter — it is itself a family, indexed by a function rather than a number, and the five names are five points in an infinite-dimensional space.

Put together, the two essays say that the taxonomy cylinders, cones and planes describes is a two-level structure: one number picks the developable surface, and one function picks the member within it. The names in a textbook are a sparse sample of the second level, chosen historically.

The whole family, audited the way every projection here is audited #

The two exact members are exact and the other three are not, and the site’s founding measurement says so without being told which is which.

What can be asked once the family is a function space #

Two things become askable that are not askable of a list.

Which member minimises a stated criterion over a stated cap? That is now a variational problem in one function rather than a choice among five, and Chebyshev’s theorem answers a special case of it: among the conformal members there is only one, so the theorem’s answer is forced. Among all members it is open, which is what every projection minimises something means when the objective is not one of the exact conditions.

What is between two members? The average of two azimuthal projections is another azimuthal projection, because averaging two radial functions gives a radial function — which is a much stronger closure property than the average of two projections finds in general, where an average of two projections from different families is not in either.

That closure has a consequence worth stating. Any convex combination of the five stays inside the family, so the whole convex hull is available and every member of it is an azimuthal projection with $f(\rho)\to\rho$ at the centre. The named five are the vertices of a region, not five isolated points.

The two exact members drawn as maps #

The reduction is a statement about functions and the projections it describes are maps, so it is worth seeing the two solutions of the two conditions as the things they are.

Reading the spacing of the parallels as the graph of $f$ is the whole reduction in a picture: four maps that differ in one curve.

The same reduction reaches the distortion. Draw the stereographic member’s indicatrices and every one of them is a circle, at every distance from the centre, which is conformality; and their sizes grow exactly as $f'(\rho)$ — the derivative of the one curve this essay has been reading. The ellipses on an azimuthal projection are a plot of that derivative and nothing else.

What the reduction does not cover #

An off-centre aspect. Everything here assumes the projection is centred where the measurement is taken from, which is what makes the reduction one-dimensional. An azimuthal projection centred elsewhere is the same function composed with a rotation of the sphere, and the site’s aspect machinery already does that — but the distortion at a point is then a function of the distance to the centre, not of the point’s latitude, which is the whole content of the aspect is a free choice.

The ellipsoid. $f$ is a function of angular distance and there is no single angular distance on an ellipsoid: the geodesic distance from a centre is not proportional to any angle, and the family does not reduce. Every ellipsoidal azimuthal projection is a series, like every other ellipsoidal projection.

The projections that are nearly azimuthal. Hammer’s and Aitoff’s are built by applying an azimuthal projection to a halved longitude and stretching the result back, which breaks the family’s defining property — azimuths are no longer preserved — while keeping most of its behaviour. They are called pseudoazimuthal for that reason, and the reduction above does not describe them: their distortion depends on two coordinates and no function of one variable will do. Where a pseudocylindrical puts its error is the same situation in the neighbouring family.

The pole of the parameterisation. Measuring the two-dimensional distortion exactly at the antipode of the projection’s centre, or at the graticule pole, returns the parameterisation’s own singularity rather than the projection’s behaviour: the stereographic projection reads 38.94° of angular deformation at the pole while $h$ and $k$ are both exactly 2. The one-dimensional scales have no such defect, which is a small argument in the reduction’s favour.

What a reader can do with the function #

The practical use of the reduction is that it makes a projection designable rather than selectable. A requirement stated as a curve — true scale on this ring, no more than this much areal error out to that distance — is a condition on $f$, and the condition can be integrated the way the two exact ones were.

Two examples of what that buys. A projection true to scale along a chosen ring rather than at the centre is $f' = 1$ shifted, which is the azimuthal equidistant with a different origin of measurement and is what a polar chart for a fixed range wants. A projection whose areal error is bounded rather than zero is a differential inequality, which has a family of solutions rather than one, and the freedom left over can be spent on the angular error — which is the shape a compromise takes when it is designed rather than averaged.

Neither is in the library, because neither has a name, and that is the point of noticing the family is a function space: which projection is best is an incomplete question partly because the set of candidates is usually taken to be the set of names.

What the reduction does to shipping one #

A projection reduced to a single function of a single variable is a different kind of object to transmit, and the difference is large enough to be worth stating beside the design uses.

A general projection is two functions of two variables. Any numerical one — a fitted map, a mixture, an optimised compromise — has to ship as a surface: a coefficient array, a spline over a two-dimensional domain, a lookup table on a grid. The size grows as the square of the resolution and the interpolation between samples is itself a design decision.

An azimuthal projection is one function of one variable, so it ships as a curve. A few hundred samples of $f$ against radius, with a cubic interpolant, reproduces the whole projection to a tolerance a user can state — and the scale factors come from the same table, since both of them are built from $f$ and $f'$ rather than measured separately. The forward map, the inverse map, the radial scale and the tangential scale are four readings of one array.

The inverse is free, which is the part with the most consequence. $f$ is monotone for any usable member, so inverting the projection is inverting a monotone function of one variable — a bisection or a monotone interpolant on the same table — where a general numerical projection needs a two-dimensional Newton iteration with a starting guess and a failure mode.

So the family’s reduction is not only an analytical convenience. It says that an azimuthal projection designed rather than named is practical to distribute, which is not true of the numerically-defined maps elsewhere on this site, and it says why: the dimension the reduction removes is the dimension in which everything else gets expensive.

The saving is a whole dimension, and it shows up everywhere the projection is used rather than only where it is stored.

That also gives the reduction a test with teeth. Any projection claimed to be azimuthal must be reproducible from such a table to the tolerance of the sampling, and a claimed member that is not is not azimuthal — the check needs no formula and no derivation, only a sweep of radii and a comparison.

Where this ladder goes #

The families field has now taken the traditional taxonomy apart in three places: the conic parameter that contains the other two families, the pseudocylindricals’ choice of what to do at the pole, the polyconic’s refusal to be built from any one cone, and — here — the reduction of an entire family to a single function of a single variable.

What is left is the equivalent reduction for the pseudocylindricals, which are two functions of latitude rather than one function of distance, and where the conditions are partial rather than ordinary differential equations. That is a harder problem and the family is correspondingly less tidy, which is exactly why its members are so numerous.