The trade-off is two lines

Every treatment of map projections says that angles and areas cannot both be preserved. Most leave it there, which invites the reading that it is a rule of thumb or an observation about the projections people happen to have invented.

It is neither, and the derivation is two lines.

The derivation #

Tissot’s indicatrix gives two numbers at every point: the principal scale factors $a$ and $b$, the largest and smallest amounts by which the map stretches the sphere there. Both definitions are statements about them.

Conformal means angles are preserved. A map preserves angles at a point exactly when it stretches equally in every direction, so

$$a = b$$

Equal-area means area is preserved. The indicatrix ellipse has area $\pi ab$ against the circle’s $\pi$, so

$$ab = 1$$

Suppose both. Substituting the first into the second gives $a^2 = 1$, so $a = b = 1$, at every point.

A map whose principal scale factors are both exactly one everywhere preserves every distance. That is an isometry. And no isometry from the sphere to the plane exists, because Gaussian curvature is intrinsic and the two surfaces have different amounts of it.

That is the whole proof.

What the argument does and does not say #

It says the two properties are incompatible everywhere at once. It does not say they are incompatible anywhere in particular.

A projection can be both conformal and equal-area at isolated points, or along a curve. Every projection with a standard parallel is exact along it — scale is true there in both directions, so $a = b = 1$ locally. What cannot happen is for that to hold across a region of non-zero area.

It also does not say the two failures are related in size. They are independent quantities, and a projection can be badly wrong about one and exactly right about the other, which is precisely what conformal and equal-area projections respectively do.

The same two lines, for any pair of properties #

The argument is not special to conformality and equal area. It works for any two of the three properties the subject names, which is why every third one is always qualified.

Add equidistance along the meridians: the map preserves distance in the north–south direction, so $h = 1$. For a projection whose graticule stays orthogonal on the map, $h$ and $k$ are the principal scales, so:

• conformal and equidistant gives $a = b$ and $a = 1$, hence $b = 1$ — an isometry.
• equal-area and equidistant gives $ab = 1$ and $a = 1$, hence $b = 1$ — an isometry.

Three properties, three pairs, and every pair forces the object the curvature forbids. Which is why no projection is ever described simply as equidistant: it is equidistant along the meridians, or from one point, or along a standard parallel. The qualifier is not modesty. It is the only form of the property that can coexist with anything else.

The same reasoning bounds how many exact properties a projection may hold at all. Each one is a constraint on two numbers, and two independent constraints exhaust them. A projection gets one property everywhere, or one property plus a curve where a second also happens to hold, and that is the whole budget.

What the empty corner is worth #

The site demonstrates the theorem computationally as well as deriving it, and it is worth being clear about what that adds.

The derivation is a proof and needs no support. What the measurement adds is a check on the library: if some projection in it ever passed both tests, the conclusion would not be that the theorem is wrong but that the implementation is. The assertion is therefore a test of the site’s own machinery, framed as a test of geometry.

That framing matters because the same measurement is what caught Web Mercator. A loop that runs every projection against both properties finds anything inconsistent, whether the inconsistency is in a projection’s claim or in the code measuring it, and it is not necessary to know in advance which one is being looked for.

The shape of the compromise #

Since both cannot be had, every projection is somewhere on a spectrum, and the spectrum has structure worth knowing.

Read together, those two figures are the trade-off drawn twice. A curve that is flat in the first is steep in the second and the reverse. Nothing is flat in both, and the shaded corner in the hero figure is where a curve flat in both would have to live.

Three ways out that do not work #

The impossibility is robust, and it is worth disposing of the obvious escapes.

Use a better projection. No. The argument does not mention any particular projection, only $a$ and $b$.

Use a smaller region. This helps in practice and does not evade the theorem. A small region has little total curvature and so admits small distortion, but the distortion is not zero and cannot be. A street map is fine because the error is below the width of the ink, not because it is absent.

Interrupt the map. Cutting the sphere into gores and laying them out separately is a real technique — Goode’s homolosine does it, and so does every globe gore printed for assembly. It reduces the distortion within each piece by making each piece small, at the cost of tearing the map apart. The pieces still distort, and now the map has gaps.

The last one is the most interesting because it identifies what is actually being traded. A projection is a continuous map of the whole sphere onto a region of the plane, and giving up continuity buys real improvements. That is why interrupted projections exist, and why they are used for thematic maps of land and never for navigation.

What the trade-off is not about #

One clarification, because it is the source of a great deal of argument elsewhere.

The theorem says nothing about which properties are worth having. It says both cannot be had at once, and stops. Whether a particular map should keep angles or areas is a question about what the map is for, and no amount of geometry answers it.

The Mercator–Peters argument is conducted almost entirely as though the geometry had a view. It does not. Both projections are exactly what they claim; each sacrifices what the other keeps; and the disagreement is about purposes, which is a legitimate disagreement conducted in the wrong vocabulary.

The one property everything can have #

There is a third classical property, and it behaves differently from the other two in a way that is worth understanding.

Equidistance means distances are preserved — but only along particular lines, or from particular points. No projection preserves all distances, since that would be the isometry the theorem forbids.

The azimuthal equidistant projection preserves distances from its centre, in every direction. The equirectangular projection preserves distances along meridians. Several conics preserve them along their standard parallels.

So equidistance is not a global property at all; it is a family of local guarantees, and a projection can offer one of them alongside neither, either, or — importantly — not alongside conformality or equal-area at the same point in the same generality.

That is why the classical taxonomy lists three properties and only two of them appear in the trade-off argument above. The third is a different kind of claim.

The azimuthal equidistant projection is the clearest instance: every distance from its centre is exact and no other distance is, which is a genuine and useful guarantee of a quite different shape from the two this essay is about.

Local exactness is always available #

The theorem forbids exactness over a region. It permits exactness on a set of measure zero, and every projection in practical use exploits this.

A tangent construction is exact along one line; a secant construction along two. Choosing where those lines fall is the main tuning available once the family and property are fixed, and it is what converts a projection from a mathematical object into a fit-for-purpose map.

The distortion then grows away from the exact lines, and for a region of modest extent it can be kept small everywhere. A UTM zone is six degrees wide and its scale error stays within about a part in a thousand — which is the curvature integral again: a narrow zone carries little total curvature, so little distortion is forced.

The result is stronger than it looks #

One more consequence, because it disposes of a hope that recurs.

The argument shows that a projection cannot be conformal and equal-area over a region. It does not assume the projection is smooth in any strong sense, or that it is defined on the whole sphere, or that it belongs to any family. It uses only that the map has a derivative with principal scales at each point.

So the prohibition survives interruption, survives piecewise definition, and survives any amount of cleverness in the construction. A projection assembled from a thousand different formulae on a thousand different patches still has principal scales at each point, and the same two lines apply at each of them.

The only genuine escape is to give up being a map of a region at all — which is what a globe does, and what an interactive rescaling tool does, and neither is a projection.

What a globe does instead #

The one object with no distortion at all is worth mentioning, since it is the alternative the theorem leaves open.

A globe is not a projection. It is the sphere, at a smaller scale, and scaling is a similarity rather than an isometry — every distance is reduced by the same factor, so every angle and every area ratio is exact.

Its cost is not distortion but occlusion: half of it is facing away at any moment, and no amount of it can be seen at once. That is a genuine trade and it is a different trade from the one this essay is about.

Which identifies what a projection actually buys. It is not accuracy — a globe has more. It is simultaneity: the whole surface, visible at once, on something flat enough to print, fold, measure and carry. The distortion is the price of seeing everything together.

An orthographic projection looks like a globe and is not one. It shows a hemisphere, flattens severely towards the rim, and is neither conformal nor equal-area — every figure on this site that resembles a globe is a projection with its own distortion.

Reading the audit plot #

The hero figure repays a careful look, because its structure is the theorem’s shape.

Points along the left edge are conformal: angular deformation at the noise floor, areal error running from small to enormous. Points along the bottom are equal-area: areal error at noise, angular deformation from moderate to severe.

Points in the middle are compromises, and they are neither close to either edge nor especially far from both.

The corner is empty. Not sparse — empty, and the two lines at the top of this essay say it will stay that way.

The one red point is Web Mercator, which is not near the left edge despite claiming to be, and that is the site’s opening argument in one mark.

Stereographic is the worked example of the left edge. Every one of its indicatrices is a circle, which is what puts it there, and they grow without bound towards the antipode, which is how far up the vertical axis it sits.

The proof needs almost nothing #

Worth noting what the argument does not assume, because that is what makes it robust.

It does not assume the projection is given by a formula, or that it is defined on the whole sphere, or that it belongs to any family, or that it is constructed geometrically. It needs only that the map has a derivative at each point, so that principal scale factors exist there.

Which means it applies to a projection assembled piecewise from a hundred different rules, to one defined by a lookup table, and to one nobody has thought of. The two lines run at every point independently, and a map is conformal and equal-area over a region only if they run successfully at every point of it.

That generality is why the result is a theorem about the sphere rather than a survey of known projections.

Six projections drawn from three different construction families — three conformal, three equal-area — are each exactly what they claim, none is both, and the theorem applies to every one of them identically regardless of how it was built.

The trade-off is stated for infinitesimal quantities — the two principal scale factors at a point. At any finite size a third failure appears that neither half of the trade-off constrains.

The trade-off is stated for infinitesimal quantities, and at any finite size a third failure appears that neither half of it constrains. A triangle drawn on Mollweide has angles that sum to 180° because the drawn triangle is a plane triangle, while the real one sums to more — so the three vertex errors have to account for the whole spherical excess between them, and no projection can make that budget smaller.

What was computed here #

The audit runs every projection in the library — twenty-one of them — through both tests, sampling several hundred points across each. The tolerances are set from measurement rather than taste: the genuinely conformal projections measure around $1.5\times10^{-6}$ degrees of angular deformation, which is the noise floor of the differentiation, and the tolerance sits sixty times above that and nearly four thousand times below the smallest real failure.

Three assertions guard the conclusion. Nothing may pass both tests. The audit must find projections of both kinds, so that a bug making everything fail would be caught. And the tolerance itself must discriminate, with the margins above and below checked on every build.

The equal-area projections measure their areal error between $6\times10^{-12}$ and $1.5\times10^{-9}$, which is a much cleaner zero than the conformal ones manage — an areal factor is a determinant and does not suffer the cancellation that comparing two nearly equal scale factors does.

What the pictures cannot show #

The audit’s axes are logarithmic and span seventeen decades, which is the only way to get a noise floor of $10^{-11}$ and a failure of $10^{4}$ onto one plot. That compresses the interesting region and exaggerates the uninteresting one, and a reader looking for a sense of how much worse one projection is than another should read the numbers rather than the distances.

The empty corner is also a picture of an absence, and an absence looks the same whether it is a theorem or an oversight. The figure cannot distinguish those; the derivation at the top of this essay is what does.

The trade, as a choice between two tilings #

The two lines are an argument about what a projection can be. There is a version of it in which the choice was actually made, by engineers, for a reason that turns out not to be the operative one.

Every screen map is a quadtree over a square projected world, and the usual defence of its projection is that a square world requires Mercator. It does not: the cylindrical equal-area with standard parallels at ±55.654° is square too. What Mercator supplies is conformality — a = b — and giving it up costs 13.8° of angular deformation at 60° north, an anisotropic pixel, and a symbol that is a different shape of ground at every latitude.

Measured at 75° north the two candidate worlds are the two lines exactly: one holds $a = b$ and lets $ab$ run to 14.9, the other holds $ab = 1$ and lets the shear reach 81°. The same forced choice shows up across four planes and three operations, where no candidate is exact in two columns — because one that was would have $a = b = 1$ everywhere, which is the isometry the theorem forbids.

Who found it, and when #

The incompatibility was understood well before it was proved. Practical cartographers knew from experience that a projection good for shape was bad for area, and the two families were named and used for centuries on that basis.

Gauss supplied the reason in 1827 with the Theorema Egregium, though he was working on the geodetic survey of Hanover rather than on map projections as such. Tissot gave the local analysis in 1859 and 1881 that turns the general prohibition into a computable quantity at each point.

Lambert had already, in 1772, produced both a conformal conic and an equal-area cylindrical projection in the same publication — and named the properties as the design goals of each. That the same person constructed both, deliberately, half a century before the theorem, is a fair indication of how well the trade-off was understood in practice before anyone could say why it held.

That is the ordinary contribution of an impossibility result, and it is easy to undervalue because nothing visible changes on the day it lands.

Lambert’s position is worth holding onto as the honest description of what the theorem added. He knew the trade existed, he knew which side each of his constructions took, and he could not have said whether some cleverer construction might have both. The proof does not improve a practitioner’s maps; it closes a search nobody could otherwise have stopped conducting.

Where this goes next #

The curvature underneath the argument is no map is faithful. The tool that makes it measurable is Tissot’s indicatrix. And for what to do about a trade-off that cannot be escaped, which projection is best.