Conformal does not mean the angles are right

Conformal is usually glossed as angle-preserving, and the gloss is exactly true and routinely misread.

What the word means #

A projection is conformal at a point if the angle between any two curves through that point is the same on the map as on the ground.

Every word of that survives measurement. Take two geodesics leaving the northern vertex above, follow each a short way, project, and measure the angle between the two image curves at the vertex: on Mercator it is the true angle to within $2\times10^{-4}$ degrees, which is the noise floor of the finite differencing rather than a real discrepancy. On the stereographic it is $6.7\times10^{-5}$ degrees, on the Lambert conformal conic $4.9\times10^{-5}$.

And the measurement discriminates, which is what makes it a measurement. The same reading on projections that are not conformal:

A factor of two hundred thousand between the two groups. Conformality is real, it is exactly what it says, and the projections that claim it have it.

What the word does not mean #

The angle between two curves is not the angle between two straight lines drawn on the map between the same endpoints, and a ruler draws straight lines.

Measured on the same triangle:

The northern vertex is out by 25 degrees, on the projection whose defining property is that it preserves angles.

Nothing has gone wrong. The two geodesics leaving that vertex are drawn as curves on Mercator, and the angle between the curves is exactly right; the angle between the chords replacing them is a different angle, and no property of the projection was ever a claim about it.

The error is fixed before the projection is chosen #

Here is the part that makes this a theorem rather than a caution.

The three angles of the drawn triangle sum to 180°, exactly, because it is a triangle in a plane. Measured across eight projections, the sum is 180.000000° every time.

The three angles of the real triangle sum to more than 180° — to 205.0154°, in this case — because a spherical triangle’s excess is its area divided by $R^2$. Measured through eight different projections, the excess comes out at 25.0154° every time, which it must, since it is a property of the triangle rather than of any map.

So the three vertex errors must sum to exactly minus the excess:

$$\sum_i \left(\text{chord}_i - \text{true}_i\right) = 180° - (180° + \varepsilon) = -\varepsilon$$

For the Mercator row above: $-0.14 + 0.17 - 25.04 = -25.01$. It cannot be otherwise, on any projection at all.

That is a conservation law and it disposes of the whole question. A cartographer cannot choose a projection that draws a large triangle with the right angles, because 25 degrees of error has to go somewhere and every flat map has 180 degrees to distribute. The only freedom is how the error is shared out among the three vertices.

How the projections differ, which is only in the sharing #

The best of them concentrates 11 degrees at its worst vertex and the worst 32, against a total of 25 that all of them must account for. A projection can spread the error nearly evenly across the three vertices, which is what the conic does here, or dump nearly all of it on one, which is what Mercator does.

Two observations follow that a reader might not expect.

The conformal projections are not the best at this. The Lambert conformal conic is best and Mercator is second worst; the stereographic is in the middle. Conformality is irrelevant to the finite question, and the ordering above is essentially a statement about how well each projection’s shape over this particular triangle happens to match a plane.

A projection can be worse than the total. Gall–Peters’ worst vertex is 32°, larger than the 25° that has to be accounted for, because it puts more than the excess on one vertex and compensates with an error of the opposite sign at another.

The best any map could do, which Legendre wrote down in 1787 #

The table of worst vertex errors ranks eight projections and has no bottom row, and the bottom row is available exactly.

The three errors must sum to −ε, so the largest of them in magnitude is at least ε/3, with equality when all three are equal. For this triangle ε is 25.0154°, so

$$\text{no projection can put less than } \frac{25.0154°}{3} = 8.34° \text{ on its worst vertex.}$$

Against that bound the ranking reads differently. The Lambert conformal conic’s 11.03° is 1.32 times the best conceivable; Mercator’s 25.04° is 3.00 times it, which is to say Mercator puts essentially the whole budget on one vertex and nothing on the others; and Gall–Peters’ 32.10° is 3.85 times it, worse than doing nothing at all.

And the projection that achieves the bound exactly is not a projection. Distributing the excess evenly over the three vertices is precisely Legendre’s theorem — reduce each angle by a third of the excess and solve the resulting plane triangle — which is a computation applied to a triangle rather than a map applied to a region. The optimum is available to a surveyor and not to a cartographer, because a surveyor treats each triangle separately and a map has to be one function over the whole sphere at once.

That is the sharpest way to state what the conservation law leaves free. The budget is fixed at ε; the best distribution is ε/3 at each vertex; a projection is a single global rule and cannot hit the optimum for every triangle simultaneously; and how close it comes for a given triangle is entirely a matter of how well its own shape happens to match that triangle’s placement. A projection’s score on this measure is a coincidence of geography rather than a property anybody designed, which is why the ordering has no relation to conformality and no relation to anything else in the library’s usual tables.

The one general statement that survives is a floor rather than a ranking: every flat map of every large triangle is wrong at some vertex by at least a third of the enclosed area over R², and the only way to do better is not to draw the triangle.

Where the misreading does damage #

Three places, in increasing order of consequence.

Reading a bearing between two distant points off a conformal map with a protractor. This is the classic error and it is the one Mercator’s whole reputation invites: on Mercator a rhumb line is straight and its bearing is exactly right, which is a genuinely useful property, and it is not the same as the bearing of the great circle. Measuring the angle at a vertex between two straight lines is measuring the rhumb bearings, which are not the geodesic bearings, and the difference over long distances is the same tens of degrees.

Computing an angle from projected coordinates in a survey. Exactly the same error at a much smaller scale, and the correction has a name — the arc-to-chord correction — precisely because the profession that could not afford the mistake worked it out. A survey applies it as a matter of routine, in arcseconds; a reader of a world map does not, in degrees.

Believing a conformal projection is the safe default. Which projection is best is an incomplete question, and “the conformal one” is an incomplete answer to it for the same reason. It is the safe default for anything infinitesimal — shapes of small features, local angles, the direction of a coastline — and it offers nothing at all for large triangles, distances, areas or the sum of anything. The word “conformal” carries an implicit locally that is dropped in almost every informal statement.

The scale at which it stops mattering #

The excess is the area divided by $R^2$, so it falls quadratically with the size of the triangle. That gives the boundary directly:

Below about 100 km the whole effect is inside the reading precision of a theodolite, which is the same threshold the plane-survey limit gives for the scale factor. Above 1,000 km it is unignorable on any map at any purpose.

So the honest short version is: conformal means the angles are right for anything small enough that everything else is also right. The property buys precisely nothing at the scale where it would be useful.

A smaller triangle, and the same law #

The conservation holds at every size, and a triangle small enough to look reasonable on a map is the case worth seeing, because the law is easier to disbelieve when the numbers are small.

That is worth stating against the intuition it contradicts. A reader shown the 25-degree case might conclude the effect is a curiosity of very large triangles, and it is not: it is present at every scale, with a magnitude equal to the enclosed area divided by $R^2$, and the only thing that changes is whether a protractor can see it.

The same observation, run the other way, is what makes the whole business tractable for a surveyor: below a certain size the excess is smaller than the instrument’s precision, so the plane triangle is the answer, and above it the excess must be distributed. There is no regime in which the excess is present and can be ignored — only a regime in which it is present and cannot be measured.

The audit ladder’s version of the same point #

This site’s headline is that Web Mercator is not conformal — it applies spherical formulae to ellipsoidal latitudes, and the measured angular deformation is 0.3848° against a noise floor of $1.6\times10^{-6}$.

Set that beside the numbers here and the proportions become clear. Web Mercator’s failure of conformality costs 0.38 degrees of angular deformation at a point. Drawing a large triangle on a correct Mercator with a ruler costs 25 degrees. The property everyone argues about is worth about 1.5% of the error nobody mentions.

That is not an argument for tolerating Web Mercator’s defect, which is a projection failing the property in its own name and misreporting it. It is an argument about which errors a reader should be told about, and the finite-angle error is two orders of magnitude larger and is never mentioned at all.

Why the confusion is so durable #

The gloss survives because it is true in the sense the person saying it usually means, and false in the sense the person hearing it usually takes.

A cartographer says conformal and means: shapes of small features are preserved, a coastline’s local wiggles come out right, a circle on the ground is a circle on the map. All of that is exactly true and is what Tissot’s indicatrix shows at a point — the circle staying a circle is precisely the statement.

A reader hears angle-preserving and takes it as a licence to measure angles, which is a statement about pairs of distant points. The word does not carry the scale at which it applies, and English has no comfortable way to say “at a point” without sounding pedantic.

The result is one of the two commonest confusions in the subject, and it is worth putting beside the other. The scale of a map is quoted as a single number and is true along at most a couple of lines; the conformality of a map is quoted as a property and is true only in the limit. Both are cases of a local statement being read globally, and both would be fixed by the same discipline this whole site is an argument for: state what was measured, and over what.

The same triangle on the projection that shares the error out most evenly, which is the best any map can do rather than a property of conformality.

What was computed here #

Each vertex carries three angles. The true angle is the difference between two initial bearings on the sphere. The tangent angle is the angle on the page between the images of the two geodesics, taken by stepping a short distance along each and Richardson-extrapolating over two step sizes to remove the leading curvature term. The chord angle is the angle between the two straight sides of the drawn triangle.

Four claims are asserted on every build. A conformal projection reproduces the tangent angle to better than $10^{-3}$ degrees, and the site’s three conformal projections do so at $2\times10^{-4}$ or better. A non-conformal projection must fail that reading by more than a degree, which the five others do by 8.7° to 38.7°. Every projection’s chord angles sum to 180° to within $10^{-9}$ degrees. And every projection’s spherical excess agrees to within $10^{-9}$ degrees, which is what makes the excess a property of the triangle.

A fifth requires every projection’s worst chord error to exceed 0.1°, so that a bug producing a triangle with the correct angles — which would mean the excess had vanished — is rejected rather than celebrated.

What the pictures cannot show #

The tangent angle is not drawable at map scale. The two curves and their chords leave each vertex within a degree or two of each other over the first few pixels, and the whole difference accumulates over the length of the side. What the figure shows is the result — two visibly different lines — rather than the angle at the vertex the argument turns on.

The excess is also invisible. A triangle summing to 205° looks exactly like a triangle summing to 180° when both are drawn with straight sides, which is the entire problem in one sentence.

The refusal that keeps this honest #

The measurement’s fifth assertion is the one worth explaining, because it is the one that would catch a plausible bug.

It requires every projection’s worst chord error to exceed 0.1 degrees. That reads backwards — a check that fails when a map is too accurate — and it is there because the failure mode it guards against is silent.

If the chord angles were computed wrongly, in a way that happened to reproduce the true angles, the drawn triangle’s angles would sum to more than 180° and the excess would appear to have vanished. Every other check would pass: the tangent readings would be unaffected, the conformal projections would still be conformal, and the essay’s tables would be full of small reassuring numbers. What would have gone is the whole subject.

So the gate requires the error to be there, at every vertex set it is given, on every projection. That is the same shape as the site’s founding rule — Web Mercator must fail conformality, Mercator must fail equal area — applied to a quantity that has no name and no advocate.

The bound also explains why Legendre’s theorem is stated for small triangles and used without apology. Its even split is optimal for the minimax, and its error — the difference between the true triangle and the plane one with reduced angles — is of higher order in the excess, so on a survey figure it is below the observations’ own noise. The theorem is not an approximation chosen for convenience; it is the exact optimum plus a term nobody can measure.

Who found it, and when #

Legendre’s theorem of 1787 is the practical form of all of this, and it is worth stating because it is the eighteenth century’s answer to the same question: a small spherical triangle can be solved as a plane triangle if each of its angles is first reduced by one third of the spherical excess. That is the conservation law above, used constructively — the excess has to be distributed, so distribute it evenly and the resulting plane triangle has the right sides.

Legendre’s theorem was the working tool of every national triangulation for a century and a half, and it is the reason the excess was a routine quantity for surveyors long before it was a familiar one for anyone else. Gauss–Bonnet, which explains why the excess is the area, arrived forty years later.

Where this goes next #

The audit ladder has spent four rungs on claims that are made about projections and not checked. What is left on it is the claims that are not made at all — the properties a projection has that nobody thinks to state, which is where the next ground begins.