The projections that gave up being one thing

The standard taxonomy sorts projections by the developable surface they were notionally rolled from. One family in it has a name derived from cones and no cone anywhere in its construction, and it was in wide official use for a century.

The construction #

Take the parallel at latitude $\varphi$. Fit a cone tangent to the sphere along it — the cone’s apex is where the tangent to the meridian at that latitude meets the axis, at a distance $\cot\varphi$ from the point along the tangent.

Unroll that cone. The parallel becomes a circular arc of radius $\cot\varphi$, drawn at its exact true length.

Now do that for every parallel, using a different cone each time, and stack the resulting arcs along the central meridian at their true spacing.

$$x = \cot\varphi\,\sin(\lambda\sin\varphi), \qquad y = \varphi + \cot\varphi\left[1 - \cos(\lambda\sin\varphi)\right]$$

That is the American polyconic, on a unit sphere with the central meridian at zero.

Why it is not a conic projection #

The taxonomy places it under cones and nothing about it is conic.

A conic projection unrolls one cone. Its parallels are concentric arcs sharing a single centre, and the whole map lies on a surface that can be laid flat. The polyconic’s parallels are arcs of different radii with different centres, all lying on the central meridian — so they are not concentric, they do not share a surface, and there is no cone the finished map can be wrapped back onto.

The construction is a limit of infinitely many cones, none of which is unrolled as a whole. Calling it polyconic is accurate about where the idea came from and misleading about what the object is, which makes it the clearest available demonstration that the developable-surface taxonomy describes an origin rather than a structure.

The family has relatives that make the point sharper. The polycylindric and polyazimuthal constructions exist by the same logic and are essentially unused, which suggests the polyconic survived on its merits rather than on its place in a classification.

What it actually preserves #

The usual tests report nothing. The site measures the polyconic at 87.9° of maximum angular deformation and an areal error reaching 4.5 — not conformal, not equal-area, and the gate requires it to fail both.

It has an exact property all the same, and neither standard test looks for it.

Every parallel is drawn at its true length. Not approximately: exactly. The parallel through $\varphi$ becomes an arc of radius $\cot\varphi$ subtending an angle $\lambda\sin\varphi$, so its drawn length is $\cot\varphi \cdot \lambda\sin\varphi = \lambda\cos\varphi$, which is the ground length of that segment of parallel.

The central meridian is at true scale, by the stacking construction.

The site asserts both by measuring the projection’s own scale factors: the parallel scale $k$ must equal one everywhere, and the meridian scale $h$ must equal one on the central meridian. Both come out at 2.4×10⁻¹² and 2.9×10⁻¹² respectively.

So this is a projection whose whole design is scale distortion — the third failure, the one the two standard measures do not report — driven to zero along an entire family of curves. Judged by conformality and equal area it has nothing; judged by what it was built for it is exact.

Measured along a meridian well east of centre, both standard distortions grow steadily away from the central meridian — which is what a projection with no global property looks like, and neither curve shows the property the projection actually has.

What it was for #

The property makes sense once the working method is known, and the working method is a nineteenth-century survey office.

The polyconic was adopted by the US Coast Survey in 1820 and used for its topographic sheets for over a century. A sheet covers a small quadrangle — a degree or half a degree on a side — and within that patch the distortion is tiny, because the central meridian passes through it and every parallel on it is at true length.

Constructing such a sheet by hand needs only a table of arc radii and spacings, and the arcs can be struck with a beam compass. That is a real advantage in an era when every sheet was drawn by a person, and it is why the projection spread: the construction is a table lookup and two compass settings.

For a single sheet the polyconic is excellent. The trouble is what happens with more than one.

The sheets do not fit #

The defect that eventually retired it, and it is a good example of a design failure that only appears at the second unit.

Each sheet is drawn about its own central meridian, so each is exact in its own middle. Place two adjacent sheets side by side and their shared edge has been drawn twice, from two different centres, with two different curvatures.

The edges do not coincide. Laid out as a mosaic, the sheets overlap in some places and gape in others, and the mismatch grows as the mosaic grows. There is no way to assemble a polyconic map of a large area from polyconic sheets.

That is a striking kind of failure. Every individual sheet is more accurate than a sheet of the same area in almost any alternative projection, and the collection is unusable. The projection optimises the unit and not the assembly, and nobody noticed until the sheets existed in quantity.

The remedy, when it came, was the transverse Mercator zone system: a projection that is worse on any single sheet and continuous across a whole zone, so sheets tile exactly. The US Coast and Geodetic Survey moved to state plane coordinates on Lambert conformal conic and transverse Mercator in the 1930s, and the polyconic went out of use for new work.

The generalisation #

There is a family of related answers, and the differences among them are instructive about what the polyconic gave up.

The rectangular polyconic modifies the spacing so that the meridians cross the parallels at right angles, sacrificing true parallel length. It was used by the British War Office.

The modified polyconic — Lallemand’s, adopted for the International Map of the World in 1909 — makes two meridians either side of the centre true to scale rather than one, and does not preserve the parallels’ length exactly. That is the secant trade applied to a polyconic, and it exists because the sheets of the International Map had to join.

Both give up the exact property in exchange for something about how sheets relate to each other. That is the tell: the exact property was the wrong thing to optimise once maps were assemblies rather than single sheets, and every successor traded it away.

In the audit it lands well out on both axes, in the company of the projections that have chosen nothing. Its actual property — exact true-scale parallels — is not an axis of that plot at all, which is the whole difficulty with placing it.

What “true to scale along a parallel” is worth #

The property sounds narrow and it is worth pricing, because a nineteenth-century survey office was not preserving parallel length out of tidiness.

A topographic sheet is used to measure. The two operations a field party performs on one are reading a position and scaling a distance, and both are done with a ruler against the sheet’s own graticule.

On a polyconic sheet, a distance measured along a parallel is correct without any correction at all — the arc drawn is the ground length. A distance measured along the central meridian is likewise correct. So the two directions a rectangular quadrangle sheet is naturally measured in are both exact, and the errors appear only for oblique lines and only away from the centre.

Set that against a conformal sheet, where nothing is exactly true to scale but everything is uniformly wrong by a factor near one. The conformal sheet needs a scale-factor correction applied to every measurement; the polyconic sheet needs none along its axes and a larger one off them.

Which of those is preferable depends entirely on whether the person using the sheet has a table of corrections and the time to apply them. In 1820 they did not, and the polyconic’s answer was the right one. By 1930 the computation had moved indoors and the assembly problem dominated, and it was not.

That is the same shape as every other choice in this subject: the projection is downstream of a working method, and when the working method changes the projection does too.

The three constructions that share the idea #

Placing the polyconic among its relatives makes the taxonomy’s difficulty concrete rather than rhetorical.

One cone, unrolled — the ordinary conic projections. Concentric parallels, one surface, and the map can be wrapped back onto the cone it came from.

One cone per parallel, none unrolled — the polyconic. Non-concentric parallels, no surface, and the construction is a limit rather than an operation on a shape.

One projection per lobe, cut apart — an interrupted projection. Several surfaces, declared discontinuities, and the pieces do not join by design.

All three are called conic, polyconic and interrupted respectively, and only the first names a developable surface that anything is actually rolled from. The second names a surface that appears in the derivation and never in the result; the third names an operation on the output rather than a construction at all.

What the family names are for #

The polyconic makes an argument the taxonomy essay could only assert, so it is worth restating with this case in hand.

The construction families predict the shape of the graticule and nothing else. The polyconic’s graticule is a straight central meridian with circular parallels and curved meridians, and knowing it is called polyconic predicts exactly that and no more.

They do not predict the property, because several families contain projections of every kind and this one contains a projection whose property is not in the vocabulary at all.

They do not predict the assembly behaviour, which is what actually retired this projection, and which no taxonomy of construction surfaces mentions.

So the useful description of the polyconic is not “a conic projection of a particular kind” but: true to scale along every parallel and along the central meridian, neither conformal nor equal-area, excellent per sheet, does not tile. That is four clauses instead of one word, and every clause answers a question somebody has.

How the failure was found #

The manner of the discovery is worth a paragraph, because it says something about which errors a design review catches.

Nothing about a single polyconic sheet is wrong. Every measurement on it is more accurate than the same measurement would be on almost any alternative, and a reviewer checking a sheet against the ground would find nothing to complain about.

The failure lives entirely in the relationship between sheets, and a relationship between sheets is not a property either sheet has. It becomes visible only when two are physically laid side by side — which happens in a map library, or in a field office assembling a mosaic, and not at a drawing board where sheets are produced one at a time.

So this is a defect that a per-unit test cannot find, no matter how thorough, because the quantity that fails is not defined on a unit. The only test that finds it is one that composes two units and checks the join, and such a test exists only if somebody anticipated that composing was going to happen.

That generalises well past cartography, and it is the reason this essay is on a site about verification. Every check on this site is a statement about one figure or one projection; a defect in how two of them relate would pass all of them. The interruption schemes are checked for their tears precisely because that lesson is available — the discontinuity is measured rather than assumed absent.

Where the idea survives #

The projection is out of use and the construction is not, which is worth a paragraph.

Any scheme that gives each piece of a map its own locally-optimal projection and accepts discontinuity between pieces is the polyconic idea. Interrupted projections are the explicit version — lobes with their own central meridians, cut deliberately. The UTM zone system is a disciplined version: sixty pieces, each with its own projection, with the discontinuity moved to boundaries chosen in advance rather than emerging between sheets.

The difference is that both of those declare the discontinuity and place it. The polyconic’s discontinuity was not declared, not placed, and not noticed until sheets were laid next to each other.

Which is the transferable lesson. A design optimised for the unit, with no statement about how units compose, has a failure mode that testing the unit cannot find — and the failure is not in any single piece.

Every parallel true to scale is a statement about the scale along the parallels, and the parallels are not the directions the polyconic stretches most.

Its directions of maximum stretch make the mismatch concrete. On the central meridian they lie along the graticule, because there the graticule is orthogonal; away from it they rotate by up to 24.5°, so the projection’s promise and its distortion are about different directions.

What was computed here #

The forward map is the standard spherical polyconic, with the $1 - \cos E$ term written as $2\sin^2(E/2)$ because the direct form loses seven significant digits for the small angles a near-equatorial parallel produces. The inverse is solved numerically by the site’s shared two-dimensional Newton method and round-trips to zero.

The projection’s exact property is asserted from its own derivatives rather than from the construction. The parallel scale factor $k$ must equal one at every sampled point from 75° south to 75° north and out to 30° from the central meridian, and it does to 2.4×10⁻¹². The meridian scale $h$ must equal one on the central meridian, and it does to 2.9×10⁻¹².

That is a stronger check than measuring a drawn arc’s length, which was the first version and failed at a part in ten million — the failure was the polyline approximation to a circular arc rather than the projection, and a discretised curve is always slightly shorter than the curve. Reading the scale factor from the Jacobian avoids the discretisation entirely.

The projection is also required to fail both standard tests, which is the check that keeps this essay’s framing honest: a projection whose only property is invisible to the two usual measures has to be demonstrably invisible to them.

What the pictures cannot show #

The sheets not fitting. That failure appears only when two sheets drawn about different central meridians are laid side by side, and every figure here is one map about one centre. Drawing the mismatch would need two separate projections in one picture with their shared edge doubled, which is a diagram of a bookkeeping problem rather than of a projection.

The figures also cannot show the property the projection has. True parallel length is a statement about the drawn length of a curve, and a drawn curve looks the same whatever its length means — which is precisely why the site reads it from the derivative and prints the number.

Who found it, and when #

Ferdinand Rudolph Hassler, the first Superintendent of the US Coast Survey, introduced the polyconic in 1820 for the survey of the American coast. Hassler was Swiss, had trained in the European geodetic tradition, and chose a projection that could be constructed accurately by hand from tables — which was the binding constraint on a survey office of that period.

It became the standard projection for US topographic mapping and was used by the Geological Survey for its quadrangle sheets into the twentieth century. Lallemand’s modified version was adopted in 1909 for the International Map of the World, the first attempt at a uniform world map series at a common scale.

Both were displaced by conformal projections on zone systems in the 1930s and 1940s, for the same reason in both cases: a map that had to be computed on and assembled beat a map that was easy to draw and exact on its own sheet.

The displacement is worth reading as a statement about what changed rather than about what was better. Hassler’s constraint was a drawing office with tables and no machines, and under that constraint a projection exact on every sheet and awkward to assemble is the right answer. The constraint that replaced it was a survey computing coordinates in bulk, and under that one a projection with a closed-form scale factor and a zone system wins outright. Neither projection improved.

Where this goes next #

The taxonomy this projection embarrasses is cylinders, cones and planes. The property it actually has is scale distortion, the third failure. And the disciplined form of its discontinuity is giving up continuity.