The conic is the whole family

The standard taxonomy names three families — cylindrical, conic, azimuthal — as though they were three things. There is one.

The parameter the atlas hides #

A conic projection is specified in practice by two standard parallels. That is the right interface for a cartographer, who knows the latitudes of the region and wants the map true along them, and it is the wrong parameterisation for the question of what a conic is.

The quantity that decides the shape of the map is the cone constant $n$: the fraction of a full turn on the page that a full turn of longitude becomes. Every conic is a fan of straight meridians radiating from a common point with concentric circular parallels crossing them, and $n$ is the only thing that distinguishes one from another once the family — equal-area, conformal, equidistant — is fixed. The two standard parallels enter only through it.

For a tangent conic touching the sphere along the parallel $\varphi_0$,

$$n = \sin\varphi_0$$

which runs from 0 at the equator to 1 at the pole. The library’s Albers projection, standard at 20° and 60°, has $n = 0.604$; its Lambert conformal conic, standard at the same pair, has $n = 0.657$. Two projections with identical standard parallels and different cone constants, because the constant is a property of the projection’s own construction rather than of the parallels chosen for it.

The two ends #

Written with $n$ in front, the family visibly has two ends, and neither of them is a conic.

At $n \to 0$ the cone has opened out into a cylinder — the surface the flattening argument shows has zero Gaussian curvature just as the cone does, which is what makes the whole family developable in the first place. Its apex has gone to infinity, the concentric parallels have become straight lines, and the fan of meridians has become a set of verticals.

At $n = 1$ the cone has closed into a plane. The apex has arrived at the pole, the parallels are circles centred on it, and the meridians radiate from it.

Substituting into the equal-area conic’s forward map makes both exact rather than suggestive. With the origin on the equator,

$$\rho = \frac{\sqrt{1 + n^2 - 2n\sin\varphi}}{n}, \qquad \theta = n\lambda$$

At $n = 1$ this is $\rho = \sqrt{2 - 2\sin\varphi} = 2\sin(c/2)$ with $c$ the co-latitude, which is the Lambert azimuthal equal-area projection in polar aspect, exactly. Expanding for small $n$ gives $x \to \lambda$ and $y \to \sin\varphi$, which is the Lambert cylindrical equal-area projection, exactly.

Measured, and at what rate #

The limits are exact statements and exact statements are checkable, so they are checked rather than repeated. The conic at each $n$ is compared with each limit point by point over a shared grid, after removing the two constants a projection is entitled to — an overall scale and an offset along $y$, since a conic’s origin is a cone apex rather than a point of the map.

Two things are worth reading off that table.

The convergence is exactly first order. Fitted across the decades, the exponent is 1.0000 towards the cylinder and 1.0000 towards the plane; for the conformal family it is 0.9999 at both ends. The departures are $1.18n$ and $1.30(1-n)$ respectively, with no quadratic term visible at any scale sampled.

First order means the conic is never nearly anything. Halving the cone constant only halves the difference from a cylindrical projection. There is no value of $n$ at which a conic is approximately cylindrical in any useful sense — at $n = 0.1$, which looks like a very open cone, the departure from the cylindrical limit is still 12% of the map’s own size. A conic is its own projection over the whole open interval, and the family’s ends are approached but never nearly reached.

That second point is the one worth having. The taxonomy’s suggestion that these are three neighbouring families understates how different they are in the middle and overstates how gradual the transition is.

The order is checked, not the values #

What the gate asserts is the exponent, fitted over six decades at each end, and the two endpoints reaching their limits to $10^{-4}$.

An exponent is a far stronger claim than a tolerance on any single value. A mistake in the normalisation, in the sampling grid or in the constant $C$ would move the coefficient and leave the exponent alone; a mistake in the geometry — the wrong relation between $n$ and the standard parallel, say — would move both. Only the right construction returns 1.0000 at both ends of both families.

The monotonicity is asserted too, at every step: the departure from the cylinder must fall as $n$ falls and the departure from the plane must rise, with no reversal. Two endpoints that happen to agree would satisfy a check on the endpoints alone.

What the numerical fit taught #

The first version of the comparison fitted an overall scale to the raw coordinates and then removed the $y$ offset. That is the wrong order, and it produced a result worth recording: the conic at $n = 0.999999$ came out as a different projection from the azimuthal one it is exactly equal to.

The reason is that a conic’s origin sits at its cone apex, 1.41 units away from the pole in these units, and a least-squares scale fitted through that offset is dominated by the offset. Centring first and fitting the scale afterwards gives the $1.3\times10^{-6}$ above; not doing so gave 1.03, which is the same order as the distance between two genuinely unrelated projections.

The check that caught it was the assertion, not the picture. A log-log plot with one bad point at the extreme left looks like a numerical artefact at the limit of resolution, which is exactly what it was not.

The conformal family does the same thing #

The conformal conic runs from Mercator at $n \to 0$ to the polar stereographic at $n = 1$. Both of those are projections with names, reputations and separate essays, and they are two members of a one-parameter family whose other members are the aeronautical charts.

That is a genuinely useful reframing. Mercator exists because a rhumb line is straight on it, and the polar stereographic exists because it is the optimum conformal map of a cap. Those two facts are usually presented as unrelated properties of unrelated projections; they are properties of the two ends of one continuum, and the aeronautical chart in the middle inherits a weakened version of each.

What the distortion does along the family #

The shape of the graticule changes continuously with $n$ and so does everything measurable about the map, which is worth seeing as a distortion pattern rather than as a drawing.

The reason to draw it that way is that the cone constant is a shape parameter and the conformality is a property parameter, and they are independent. Changing $n$ slides a projection along the family; changing family swaps which property is exact. Two knobs, and the taxonomy conflates them because it only has names for a few of the settings.

Three of the library’s members show the two knobs turning separately. An equal-area conic and a conformal conic share a construction and differ in property; an equal-area conic and an equal-area cylindrical share a property and differ only in cone constant, which is the axis this essay is about. The taxonomy has a name for each of the three and no name for the axis.

Why the taxonomy exists anyway #

The three-family classification is not wrong, and dismissing it would be a mistake of the kind the developable-surface essay is careful about — a cylinder and a cone really are developable, and the fact that they are the two ends of one continuum does not make either of them a fiction.

It is a classification of construction, and it accurately describes how the projections were originally derived and how their graticules look. What it does not do is predict any property anyone cares about: each family contains conformal members, equal-area members and compromises, so knowing that a projection is conic tells nobody whether angles or areas are preserved.

The cone constant is a better parameter for the same reason. It is continuous, it is measurable from the map itself — count the fraction of a full turn the sheet occupies — and it predicts what the map looks like. It still predicts nothing about conformality or equal area, because those are chosen independently.

The family table makes that independence explicit as a grid. The result here adds one thing to it: the rows of that grid are not three discrete cases but one axis with two endpoints, and every column runs along it.

What a standard parallel means in this language #

A standard parallel buys a line of true scale, and putting two of them on a conic is the secant construction: convert one zero line into two, with error of the opposite sign between them.

In terms of the cone constant the relation is simple and slightly surprising. A tangent conic has $n = \sin\varphi_0$ for its single standard parallel. A secant conic with parallels $\varphi_1$ and $\varphi_2$ has a cone constant that depends on the family — $(\sin\varphi_1 + \sin\varphi_2)/2$ for the equal-area case, and a logarithmic expression for the conformal one — and in both cases it lies strictly between $\sin\varphi_1$ and $\sin\varphi_2$.

Every projection minimises something, and in this language the objective is what picks the family while the cone constant is what the objective is then optimised over. So the two standard parallels of an atlas conic are a way of specifying $n$ together with a scale, and the cartographer’s interface and the geometer’s parameter are related by an invertible map. That is why the same Albers projection can be described as “standard at 20° and 60°” or as “cone constant 0.604” without either description being more fundamental — though only one of them is a single number.

The one-parameter view as a design tool #

Treating the family as a continuum turns projection selection for a mid-latitude region into a one-dimensional optimisation, which is a much smaller problem than choosing among named projections.

Given a region and a criterion, the cone constant that minimises the criterion is a single number found by a line search, and the resulting projection may well have no name. That is not a difficulty: naming a projection is not measuring it, and a nameless conic at $n = 0.71$ fitted to a specific country is a better map of it than the named one at 0.604 that happens to be in the software.

National mapping agencies have been doing exactly this for a century, which is why the world’s national grids are a list of conics with idiosyncratic standard parallels rather than a list of standard projections. The standard parallels are the fitted parameter wearing the atlas’s clothes.

What the continuum adds is the observation that the search has no interior obstacles. The family is smooth in $n$ over the whole open interval, every member is a genuine projection with the family’s property exact, and the only special points are the two ends — where the map stops being a conic and becomes something with its own name. A line search cannot get stuck, and a result at $n = 0.98$ is a signal that the region wants an azimuthal projection rather than a conic, which is a different and more useful piece of advice than a number.

One member measured across latitude rather than against its limits behaves as every member does at its own latitude: exact at the parallel its cone is tangent along, and departing either side of it.

What was computed here #

Two conic families were implemented with the cone constant as their parameter and the tangent relation $\sin\varphi_0 = n$ fixing the remaining constants. Each was compared with its two limits at nine values of $n$ spanning six decades either side, over a shared grid of 195 points, by centring both coordinate sets in $y$, fitting one scale by least squares, and taking the largest residual.

Four claims are asserted: the departure from the azimuthal limit is below $10^{-4}$ at $n = 0.999999$; the departure from the cylindrical limit is below $10^{-4}$ at $n = 10^{-6}$; both sequences are monotone at every step; and the fitted exponents are within 0.02 of one at both ends of both families.

Both new projections also pass the property their names claim, measured the ordinary way: the equal-area conic to $10^{-10}$ in areal factor and the conformal one to $1.6\times10^{-6}$ degrees of angular deformation, which is the site’s noise floor.

What the pictures cannot show #

The horizontal axis of the limits figure is a logistic scale in $n$, which compresses the middle of the family — the part an atlas actually uses — into the centimetre where the two curves cross. That is the right axis for the claim being made and the wrong one for browsing the family, which is what the panel figures are for.

The panel figures in turn show three members and imply a continuum. Nothing in a set of three pictures establishes that the intermediate cases behave; the measurement does.

Who found it, and when #

Lambert’s 1772 memoir introduced both the conformal conic and the azimuthal equal-area projection, and derived them within a few pages of each other. Whether he regarded them as members of one family is not obvious from the text; what is clear is that the machinery was in place from the beginning, since the conic’s cone constant is right there in his construction.

The explicit one-parameter treatment is a nineteenth-century tidying, and the modern habit of teaching three families is later still — a pedagogical convenience that has outlived the moment when it was easier than the alternative.

Why these three points and not three others #

The taxonomy names three families and the parameter is continuous, so the obvious question is what is special about the three values that got names. The answer is not that they are geometrically distinguished, and it is worth being precise about what they are.

They are the values where the arithmetic degenerates. At one end the cone opens out until it is a cylinder and the cone constant goes to zero; at the other it closes until it is a plane and the constant goes to one. Those are the limits of the parameter’s range, and a limit is where a formula simplifies: terms vanish, a general expression collapses to a special one, and what remains can be written on a page and evaluated by hand.

That is a fact about the era the names come from. A projection had to be computed with logarithms and a table, so a member whose formula lost two terms was not marginally more convenient — it was the difference between usable and not. The three named families are the three places on one axis where somebody could actually do the sums, and the continuum between them was known, derivable and impractical.

Nothing about the geometry marks them out. The distortion varies smoothly along the parameter with no feature at the ends; a member at a cone constant of 0.62 is not a compromise between two kinds of thing, it is simply a conic. The measurement in this essay is what says so — the intermediate cases behave, at a measured rate, and the endpoints are where the curve stops rather than where it does anything.

It is worth noticing that the same argument applies to the other names in the subject, and mostly gives the same answer: the projections with names are overwhelmingly the ones with closed forms, and the ones with closed forms are the ones somebody could evaluate before there were machines.

The constraint that produced the taxonomy is gone and the taxonomy is not, which is the ordinary fate of a good simplification. It survives because it is taught, because atlases are indexed by it, and because a name is easier to ask for than a parameter — and the cost of it surviving is that a designer thinks of the choice as which of three rather than as what value, which is a choice between three points on a line that has all of it available.

Where this goes next #

The conic family is one axis. The pseudocylindricals form another, and their defining choice is not a continuous parameter at all but a decision with two answers: what to do at the pole.