Cylinders, cones and planes

Every introduction to map projections sorts them the same way. Wrap a cylinder round the globe for cylindrical projections, fit a cone over it for conic ones, lay a plane against it for azimuthal ones. Unroll it and there is a map.

The picture is memorable and mostly a fiction, and the taxonomy it supports sorts on the wrong axis.

What the picture gets right #

The output shape, which is not nothing.

Cylindrical projections produce a rectangle, with the graticule as a right-angled grid. Meridians are equally spaced vertical lines and parallels are horizontal.

Conic projections produce a fan. Meridians radiate from a point and parallels are concentric arcs.

Azimuthal projections produce a disc. Meridians radiate from the centre and parallels are concentric circles.

These are real distinctions with real consequences, and they map onto region shapes usefully — a fan suits a country wide in longitude, a disc suits a roughly circular one.

What it gets wrong #

Two things, and the second is more serious.

Most of these projections are not obtained by projecting onto anything. Mercator’s parallel spacing is $\ln\tan(\pi/4+\varphi/2)$, which no arrangement of rays from any light source produces — it is the function that makes the projection conformal, derived from the requirement rather than from a construction. The same is true of the equal-area cylindricals, of Mollweide, of Robinson. The geometric ones are a minority.

The developable surface contributes nothing to the fidelity. A cylinder unrolls perfectly, which invites the conclusion that the wrapping is harmless and the distortion must come from somewhere else. It comes from getting the sphere’s surface onto the cylinder, which is where all the stretching happens; the unrolling afterwards is faithful and there is nothing left to save.

So the family name describes the shape of the answer, not the quality of it.

Why the mismatch matters #

The properties that decide a projection’s suitability are conformality, equal-area and the various compromises. The construction families cut across all of them.

The cylindrical family alone contains Mercator (conformal), Gall–Peters and Lambert’s cylindrical (equal-area), Miller (neither, deliberately), and the plate carrée (neither, by accident). Those four have nothing in common that a map user cares about.

Conversely the conformal projections are spread across families: Mercator is cylindrical, Lambert conformal conic is conic, stereographic is azimuthal. Someone who needs conformality gains nothing from knowing the families and everything from knowing the region’s shape.

Which is why choosing a projection starts with the property and the region rather than with the taxonomy — and why the taxonomy, taught first and remembered longest, sends people to the wrong question.

The azimuthal family is the exception #

One family does have a genuine unifying property, and it is not the one in the name.

Every azimuthal projection preserves directions from its centre. A straight line from the centre of the map to any point is a great-circle route, at the correct bearing. That is what azimuthal means, and unlike the other families it is a property rather than a shape.

It also has a consequence worth noticing: azimuthal projections are the natural choice for anything radiating from a point. Air routes from a hub, seismic waves from an epicentre, broadcast range from a transmitter.

The family still contains projections with quite different secondary properties — stereographic is conformal, Lambert azimuthal is equal-area, gnomonic is neither and straightens every great circle — so the mismatch persists even here. But the shared property is real.

The tangent-and-secant refinement #

Within each family there is a standard sub-distinction: whether the surface touches the sphere along one line or cuts through it along two.

A tangent cylinder touches at the equator, and the projection is exact there. A secant cylinder cuts at two parallels, and the projection is exact along both while being slightly compressed between them. The same distinction applies to cones, where a secant cone with two standard parallels is the usual choice.

This is a real and useful refinement, and it is about where the distortion is zero rather than about how much there is. The total distortion is not reduced by using a secant surface; it is redistributed, with two zero lines instead of one and a small error of the opposite sign between them.

What a better taxonomy would sort on #

Property first, region shape second, construction last — which is the order Snyder’s manual uses and the order the tables in most software documentation do not.

The construction is worth knowing for a different reason: it settles what the map will look like, which matters for layout, for how the graticule reads, and for whether the poles are points or lines. Those are legitimate design considerations. They are just not the ones that decide whether the map is fit for its purpose.

The pseudo- families #

Two more families appear in most taxonomies and sit awkwardly in the wrapping picture, which is a further sign the picture is doing less work than it seems.

Pseudocylindrical projections have straight parallel parallels, like cylindrical ones, and curved meridians. Sinusoidal, Mollweide, Eckert IV and Robinson are all of this kind. There is no cylinder anywhere in their construction; the name means “like a cylindrical projection, except”.

Pseudoazimuthal projections, such as Hammer, are similar departures from the azimuthal pattern.

The prefix is doing the work the taxonomy cannot. These projections are defined by properties and by the shape of their graticule, and the family names are descriptions after the fact rather than constructions.

Why the poles are the tell #

A quick way to read a projection’s family from its graticule, and the one distinction that genuinely matters visually.

Poles as lines — the whole top edge of the map is the north pole. Every cylindrical projection does this, and so do Robinson and Eckert IV. It is topologically false and keeps the high latitudes readable.

Poles as points — the meridians converge. Sinusoidal, Mollweide, Hammer and every azimuthal projection do this. It is correct and squeezes the polar regions.

That single choice affects a map’s appearance more than the family name does, and it is a design decision rather than a consequence of the construction.

What the family does predict #

Being fair to the taxonomy: it predicts several things reliably, and they are worth listing.

The graticule’s shape. Straight parallel lines, concentric arcs, or concentric circles. This is what the family name really encodes.

Where the projection is symmetric. A normal-aspect cylindrical projection is symmetric about the equator and about every meridian; a conic about its central meridian; an azimuthal about its centre.

Which regions it suits. Directly downstream of the symmetry — a fan for a region wide in longitude, a disc for a compact one.

How the poles are treated. Cylindrical projections make them lines; azimuthal projections make one a point and the other infinitely distant.

Those are useful and they are all about layout rather than about fidelity, which is the point the essay is making rather than an objection to the taxonomy existing.

Where the taxonomy came from #

The historical reason for its dominance is worth knowing, because it explains why it persists.

Before analytic methods, projections really were constructed geometrically. Drawing a projection meant literally projecting — rays, a surface, a construction with compass and straight edge — and under those conditions the developable surface is the whole content of the method.

Lambert’s 1772 treatise is the turning point: it defines projections by the properties they must have and derives the formulae from the requirement. After that the geometric construction becomes a description rather than a method, and most projections invented since have no construction at all.

The taxonomy is a fossil of the pre-analytic era, preserved because it is easy to teach and because the graticule shapes it predicts are genuinely useful.

What to teach instead #

A suggestion, since the essay is largely a complaint.

Start with the two properties and the fact that they cannot coexist. That is two lines of algebra and it is the whole structure of the subject.

Then the measurement: an indicatrix at a point, with its numbers, so that “distortion” becomes a quantity rather than an impression.

Then the purposes, and which property each requires.

The families come last, as a fact about what the finished map looks like, which is what they are. A student who has met the properties first will read the taxonomy as layout information and will not expect it to answer questions it cannot.

The current order — families first, properties later, purposes rarely — produces people who can recite that Mercator is cylindrical and cannot say what it is for.

One final defence of the taxonomy, since the essay has been dismissive. It is a good vocabulary for talking about maps: saying “a conic” conveys a shape instantly and unambiguously, and shape is what a person looking at a map notices first. The complaint is not that the words are useless but that they answer a question about appearance while being taught as though they answered one about accuracy.

There is a broader version of the complaint worth carrying elsewhere. Any taxonomy inherited from an older methodology sorts on what that methodology could see, and continues to be taught long after the methodology has gone. The cylindrical-conic-azimuthal division is a fossil of the era when projections were drawn with rays and a straight edge, and it survives because it is easy to draw on a blackboard rather than because it answers anything.

Worth noting too that the families say nothing about a projection’s domain. A cylindrical projection may cover the whole sphere or be truncated at 84°; an azimuthal one may show a hemisphere or a fraction of one. The domain is set by where the projection diverges, which follows from the formula rather than from the family, and it is often the first thing a map user needs to know.

The families are also unhelpful about the one thing a reader most often wants, which is where a given map is accurate. That is set by the standard parallels and the aspect, neither of which the family name records, and both of which are recoverable from the projection’s parameters in a line. A description reading “conic, standard parallels 29.5° and 45.5°” says something useful; “conic” alone does not.

A closing thought on why the fiction persists in teaching. The wrapping picture is physically intuitive and can be demonstrated with a sheet of paper and a ball, and no equally vivid demonstration exists for “this projection is the function whose parallel spacing makes the vertical stretch match the horizontal one”. Intuitive and wrong beats correct and abstract in a classroom, and the cost is students who can name families and not properties.

The taxonomy also predicts nothing about a projection’s aspect, which is a free parameter orthogonal to everything the families encode. A transverse Mercator and a normal Mercator are the same projection in the same family with the same property, and they look and behave completely differently — so even within a family, the name leaves the most consequential design choice unstated.

Reading the taxonomy as a description of output rather than of method also removes the puzzle about why it survives at all. It is genuinely good at what it does — telling a reader what shape of map to expect — and the complaint is only that it is asked to do something else.

The same warning applies to any classification inherited without its context. A scheme built to answer one question will keep being applied to another, quietly, for as long as nobody checks what it was for — and the check is usually the same one: ask what the categories predict, and compare that against what the reader wanted to know.

What was computed here #

The family table is generated from measurements rather than from a stored taxonomy. Each projection’s property is determined by running the definitions as computations and taking the worst case over several hundred points; the family is the only thing in the table that is declared rather than measured.

The figure carries an assertion: at least two construction families must contain projections of more than one property class. That is the figure’s whole argument, and if the families ever lined up neatly with the properties, the build would stop rather than draw a table contradicting its own caption.

What the pictures cannot show #

The wrapping, because it does not happen. A figure showing a cylinder round a globe would be illustrating the fiction rather than the projection, and this site does not draw one for that reason.

The table also flattens a genuine subtlety: “compromise” covers projections that are near-conformal, near-equal-area, and neither, and grouping them loses information that the distortion measurements carry.

Who found it, and when #

The three developable surfaces have been the organising principle since at least Lambert’s 1772 treatise, and Lambert himself constructed projections in all three families in that single publication — along with naming their properties as the design goals.

The taxonomy hardened into the standard teaching sequence during the nineteenth century, when the geometric constructions were still the dominant way of thinking about projections and the analytic ones were newer.

Snyder’s Map Projections: A Working Manual (1987) organises by property and application instead, which is a deliberate departure and the reason it remains the standard practical reference.

Where this goes next #

The surfaces the families are named after are what can be unrolled. What a standard parallel does within a family is what a standard parallel buys. And the free choice the taxonomy usually omits is the aspect.