Why Mercator exists

Mercator is the most criticised projection in cartography and one of the most successful pieces of applied mathematics ever produced. Both are true, and the second is easier to establish.

It was built to answer one question, and it answers it exactly.

The problem in 1569 #

A ship at sea has a compass. It has no way of knowing where it is beyond dead reckoning, and no practical way of continuously adjusting its heading against a curve.

What it can do is hold a bearing. Set the wheel so the compass reads a constant angle and sail. The curve that results crosses every meridian at the same angle, and it is called a rhumb line or a loxodrome.

So the navigator’s question is not “what is the shortest route” — that route cannot be sailed with the available instruments. It is: given a chart, can a course be laid off with a straight edge and read as a compass bearing?

For that to work, the map must have one property. A rhumb line crosses every meridian at a constant angle; if the map preserves angles, the rhumb line’s image crosses every projected meridian at that same constant angle; and if the meridians are parallel vertical lines, a curve crossing them all at a constant angle is straight.

Conformality plus straight parallel meridians gives straight rhumb lines. That is the specification, and Mercator’s projection is what satisfies it.

Conformality is the requirement, not the feature #

This is worth stating plainly because it is usually the other way round in popular accounts.

Mercator is not a projection that happens to be conformal and happens to be useful for navigation. It is the projection derived from the navigational requirement, and conformality is the constraint that requirement imposes. The characteristic spacing of the parallels — increasing toward the poles as $\ln\tan(\pi/4 + \varphi/2)$ — is exactly what is needed to keep the vertical stretch equal to the horizontal stretch $\sec\varphi$ that straight meridians force.

The famous areal distortion is the direct consequence. Longitude is stretched by $\sec\varphi$; conformality demands latitude be stretched by the same; and the areas therefore go as $\sec^2\varphi$, which is fifteenfold at 70°.

The distortion is not a defect of the design. It is the design, correctly executed.

The claim, checked #

The site does not take the straightness on trust.

A rhumb line is generated by integrating its defining property — stepping along the sphere holding a constant bearing, which makes no reference to any projection. That curve is then projected and its departure from the straight line through its endpoints is measured.

On Mercator, the departure is $3.6\times10^{-7}$ of the curve’s own length. And that residue is itself demonstrated to be the integrator rather than the geometry: refining the step by a factor of sixteen shrinks it by a factor of 256, which is fourth-order convergence and is what numerical error does. Geometry would not shrink.

The same rhumb lines on an equirectangular map bow by up to $10^{-2}$ — four orders of magnitude more.

What the straight line costs #

The rhumb line is longer than the great circle, and the excess is worth quantifying because it is the price of being able to steer at all.

Cape Town to London costs three kilometres — nothing, because the route is nearly along a meridian and a meridian is both a rhumb line and a great circle. New York to Madrid costs 171 km, about three per cent. London to Tokyo costs 1,737 km, eighteen per cent. Anchorage to London costs 2,031 km, twenty-eight per cent.

The pattern is that the penalty grows with latitude and with east–west extent, and vanishes for north–south routes. For the Atlantic trade that made Mercator’s chart valuable, three per cent was an acceptable price for a course that could actually be steered.

For a modern transpacific flight it is not, which is why aircraft fly great circles — and why the gnomonic projection has a job.

The standard method #

The classical solution used both projections together, and it is a nice piece of engineering.

Plot the route on a gnomonic chart, where the great circle is a straight line. Read off a series of waypoints along it. Transfer those to a Mercator chart and join them with straight segments. Each segment is now a rhumb line with a constant bearing, and the sequence of them approximates the great circle as closely as the number of waypoints allows.

Two projections, each chosen for the one property it has exactly, composed to solve a problem neither solves alone. This was standard practice for ocean navigation into the twentieth century.

What Mercator is bad at #

Everything else, and the essay would be dishonest not to say so.

Five identical patches of sphere on Mercator make the cost concrete: the one at 70° comes out fifteen times larger than the equatorial one relative to its true size. That is exactly the consequence of conformality plus straight meridians, computed rather than estimated.

As a world map for anything to do with size, area, or the relative importance of places, Mercator is a poor choice and there are many better ones. The criticism is legitimate and is the subject of its own essay.

What does not follow is that the projection is bad. It is a tool built to a specification, meeting the specification exactly, and being used for a purpose it was never designed for. The blame for a Mercator world map on a classroom wall belongs to whoever hung it there.

The modern irony #

Mercator’s straight rhumb lines are largely obsolete for navigation. GPS and autopilots make continuous course correction trivial, so a modern vessel steers great circles directly and the projection’s whole reason for existing has evaporated.

And Mercator is now more dominant than it has ever been, because it turned out to have a second property nobody was designing for: at every zoom level, on every tile, north is up and the projection is locally conformal, so a street corner looks like a street corner. For a map that will be panned and zoomed continuously and viewed mostly at city scale, that matters far more than what the projection does to Greenland.

The projection survived the disappearance of its purpose by acquiring a new one. The irony is that the version now in use is not actually conformal, so it has kept the reputation and lost the property.

The scale factor, and what it means on a chart #

A working navigator needed more from a Mercator chart than straight rhumb lines. Distances had to be measurable off it, and on a projection whose scale changes with latitude that requires a convention.

The convention is to measure distance against the latitude scale at the same latitude as the leg being measured, never against the longitude scale and never against the latitude scale elsewhere on the chart. One minute of latitude is one nautical mile, everywhere, by definition — and Mercator’s conformality means the local scale is the same in every direction, so the latitude scale at that latitude is the correct ruler for a leg in any direction there.

That is conformality earning its keep a second time. On a non-conformal chart the same trick would fail, because the scale would depend on the direction of the leg as well as its position.

Measured, its angular deformation is flat on zero — which is what lets one ruler serve every direction — and the areal factor is the price, growing as the square of the linear scale factor.

Why the poles are missing #

Mercator’s ordinate is $\ln\tan(\pi/4 + \varphi/2)$, which grows without bound as latitude approaches 90°. The poles are at infinity and no chart can include them.

That is not a defect to be patched. It is forced: conformality with straight parallel meridians requires the vertical stretch to match the horizontal stretch $\sec\varphi$, and $\sec\varphi$ diverges at the pole. Any projection with those two properties has the same behaviour.

Charts are conventionally cut at about 84° north and south, which is where this site’s library truncates too. For a navigational chart that limit costs nothing — there was no shipping there — and for a web map it is why the slippy-map tile scheme is square: cutting at about 85.05° makes the map exactly as tall as it is wide, which is a convenient accident rather than a geographic decision.

The projection is not a picture of a cylinder #

Worth saying once directly, because the standard illustration implies otherwise.

There is no light source, no cylinder, and no geometric projection that produces Mercator. Wrapping a cylinder round a globe and projecting rays from the centre gives the central cylindrical projection, which is a different and much worse map — its parallel spacing goes as $\tan\varphi$ rather than $\ln\tan(\pi/4 + \varphi/2)$, and it is not conformal.

Mercator’s spacing is the solution to a differential equation: make the vertical stretch equal the horizontal one at every latitude. That is an integral of $\sec\varphi$, and it is why the cylinder in the family name is a description of the output shape rather than a recipe.

What the chart looked like in use #

Worth picturing, because the projection’s design makes sense only against the working method.

A navigator laying a course took a straight edge, drew a line from the departure point to the destination, and measured its angle against any meridian on the chart — all meridians being parallel, any one would do. That angle was the compass course, read directly, with no computation.

Distance came from the latitude scale at the relevant latitude, one minute of latitude being one nautical mile. Position was plotted by dead reckoning: course, speed and elapsed time, marked forward along the line.

Every one of those operations depends on a property of the projection. Straight rhumbs make the ruled line a course. Parallel meridians make any of them a protractor. Conformality makes one ruler serve every direction at a given latitude.

The chart is a computational instrument, and the projection is its design.

Every one of its indicatrices is a circle, which is what lets a single latitude scale serve as a ruler for a leg in any direction: the local scale does not depend on which way the leg points.

The successors #

Two later projections solve the same problem better in narrower circumstances, and both are worth knowing.

Transverse Mercator applies the same construction about a meridian rather than the equator, which makes it exact along a north–south line. It is the basis of UTM and of most national grids, and it is the projection most of the world’s survey data actually lives in.

The oblique Mercator puts the exact line along an arbitrary great circle, which suits a country elongated on a diagonal or a satellite’s ground track.

All three are the same projection with the axis pointed somewhere else, and the family’s continued dominance in technical work is a fair measure of how good the original specification was.

Conformal projections exist in three families and Mercator’s particular virtue is not conformality, which it shares with all of them. It is conformality combined with straight parallel meridians, and that combination is what makes the rhumb lines straight.

The projection outliving its purpose #

One closing observation, because it is the neatest thing about the projection’s history.

Mercator solved a problem that no longer exists. Satellite navigation and autopilots make continuous course correction trivial, so no vessel steers a rhumb line for want of an alternative, and the property the projection was designed around is of no practical use to modern navigation.

It is nonetheless more widely used than at any time in its history, for a property nobody was designing for: at any zoom level, on any tile, north is up and shapes are locally right, which is what a panned and zoomed street map needs.

A tool built to an exact specification, surviving the disappearance of that specification by accidentally satisfying a different one. And the version now in use lacks the property it is named for, so it has kept the reputation and lost the substance.

The rhumb line’s excess is the price of being able to steer at all, and it is not the price anyone actually pays: a route is broken into legs, each of them a rhumb line, and the excess falls with the square of their number.

What was computed here #

The rhumb lines are integrated from $\mathrm{d}\varphi = \delta\cos\theta$, $\mathrm{d}\lambda = \delta\sin\theta/\cos\varphi$ with a midpoint rule, and the integration is verified by requiring the path to land on its intended destination — it does, to between $2\times10^{-11}$ and $1.4\times10^{-7}$ radians.

That independence matters. The first version of the path generator interpolated latitude and longitude linearly, which is a different curve and not a rhumb line at all; it bowed by nearly one per cent on Mercator and the straightness assertion caught it. Generating the curve from Mercator’s own ordinate would have made the test circular and would have hidden the error.

Distances use the haversine formula rather than the spherical law of cosines, which is algebraically equivalent and loses precision badly at short range. The great circle is asserted never to be longer than the rhumb, across all five journeys.

What the pictures cannot show #

A rhumb line spirals infinitely toward the pole without reaching it, which the finite journeys drawn here never approach. Mercator’s straight line reaches infinity at the pole, and the two facts are the same fact.

The projections drawn are spherical. The Earth is an ellipsoid and the real rhumb lines differ slightly, by an amount that matters for surveying and does not for the argument here — and the figures say which body they assume.

Where the property is used now #

The rhumb line was the reason and it is no longer the use. What keeps the projection universal is a second consequence of the same conformality: a map that does not shear under rotation, on which a pixel is a square of ground and a circular symbol is a circle on the Earth.

That property is what a tiling scheme is built on, and the applied field measures what the alternative would cost — 13.8° of angular deformation at 60° north for the equal-area cylindrical whose world is also square. The scheme’s other inheritance from this projection is the cut at 85.0511°, which is where the northing equals half the world’s width.

The modern use is a quadtree over the projection’s square world, each level exactly half the one above, and the ground resolution at 55° north differs from the published figure by this essay’s own $\sec\varphi$. The world has to be stopped somewhere for that to work, and the latitude is not a rounding: 85.0511287798° is where the northing equals half the world’s width, which is the only cut a single square root tile allows. The price is 1,901,487 square kilometres.

Who found it, and when #

Gerardus Mercator published the chart in 1569 with the title Nova et Aucta Orbis Terrae Descriptio ad Usum Navigantium Emendate Accommodata — a new and enlarged description of the Earth, corrected for the use of navigators. The purpose is in the title.

He did not publish a derivation, and probably did not have one: the mathematics needed is an integral of $\sec\varphi$, and calculus was a century away. He appears to have constructed the spacing geometrically and by trial.

Edward Wright supplied the mathematical justification in 1599, computing the required spacing by summing secants in small steps — a numerical integration performed by hand, decades before the operation had a name.

How good that hand computation was can be checked, and the answer is that it was better than the paper it was printed on. Summing $\sec\varphi$ at one-arcminute intervals and comparing against $\ln\tan(\pi/4+\varphi/2)$ gives an ordinate short by $1.5\times10^{-4}$ at 60° — which corresponds to placing the 60th parallel about 460 metres too far south on the ground, and about 690 metres at 75°. On a chart of the Atlantic at any scale a navigator could draw on, that is invisible.

It is also the error of the crudest possible rule. Wright’s sum evaluates the secant at the start of each interval; taking it at the midpoint instead, which costs nothing, drops the discrepancy to under a decimetre everywhere below 75°. Whether Wright did that is not recorded, and either version was already several orders of magnitude finer than a ship’s ability to know where it was.

The point is not that the arithmetic was easy. Sixty secants per degree, to seven figures, by hand, for ninety degrees, is a serious undertaking, and it was carried out to supply a justification for a chart already thirty years in use.

That last point is the one worth carrying out of the history. The chart came first and worked; the mathematics arrived thirty years later to say why. A construction that is right before anybody can justify it is the normal order in this subject rather than the exception.

Where this goes next #

The path Mercator does not straighten is the shortest route is not straight. The projection that straightens it instead is the gnomonic companion. And the argument about what Mercator costs is Mercator against Peters.