Cartographic Projection
every map is wrong, and by exactly how much
Paths and directions

The shortest route is not straight

The shortest path between two points on a sphere is an arc of a great circle, and on almost every map it is a curve. The straight line on a Mercator chart is a different route entirely, and on some journeys it is twenty-eight per cent longer.
14 min read 7 figures Computed, not quotedEvery figure here is a projectionPurpose before property

A flight from London to Tokyo goes over the Arctic. On a wall map that looks like a detour — Tokyo is to the east and the aircraft heads north — and it is the shortest route by seventeen hundred kilometres.

The map is what is misleading, and the amount by which it misleads is computable.

London to Tokyo on MercatorTwo routes. The great circle is 9559 km and is the shortest path on the sphere. The rhumb line holds a single compass bearing the whole way and is 11296 km — 1737 km further, or 18.2 per cent. On Mercator the rhumb line departs from straight by 5.0e-9 of its own length.LondonTokyosolid: shortest · dashed: constant bearing+18.2% for the rhumbdrawn in Mercator
Fig. 1 London to Tokyo on Mercator. The solid curve is the shortest path on the sphere, 9,559 km. The dashed straight line is the route of constant compass bearing, 11,296 km — 1,737 km further, or eighteen per cent.

What a great circle is

Slice a sphere with a plane through its centre and the cut is a great circle: the equator, every meridian, and infinitely many others. Any two points that are not antipodal lie on exactly one.

The arc between them is the shortest path on the surface. That is what makes it the sphere’s geodesic — the analogue of a straight line, and the curve a stretched string on a globe follows.

It has one property that makes it awkward for navigation and is the source of everything else here: its compass bearing changes continuously along the route. The London–Tokyo great circle leaves London heading north-north-east and arrives in Tokyo heading south-east, turning steadily the whole way. Nothing is steering wrongly; the meridians are converging underneath.

Why it curves on the map

Because the map is flat and the route is on a sphere, and no map can be faithful.

Slightly more precisely: a projection sends geodesics to straight lines only if it is specifically built to, and almost none are. Mercator is built to straighten the other curve — the constant-bearing rhumb line — and having straightened that, it cannot also straighten great circles, because they are different curves.

So a great circle on Mercator bows. This site measures the bow at about one per cent of the chord’s length for the journeys it draws, which is small as a fraction and enormous as a distance.

London to Tokyo, seen four waysThe same two routes on four projections. The gnomonic projection renders every great circle as an exactly straight line, which is what it is for; Mercator renders every rhumb line straight instead. Neither path changed — only the map did.Mercatorgreat circle bows 3.7e-1Gnomonicgreat circle bows not at allEquirectangulargreat circle bows 2.0e-1Orthographicgreat circle bows 9.0e-2solid: shortest · dashed: constant bearingone pair of routes
Fig. 2 The same two routes on four projections. On the gnomonic projection the great circle is exactly straight, which is the only thing that projection does and the reason it exists. On Mercator the rhumb line is exactly straight instead. Neither path changed; only the map did.

Read that figure carefully, because it contains the whole idea: straightness is a property of the map, not of the path. The same great circle is straight in one panel and curved in three, and the aircraft flies the same route in every case.

What the alternative costs

The rhumb line is sailable with a compass and is longer. How much longer depends entirely on the journey.

What a constant compass bearing costsThe extra distance of the rhumb line over the great circle, for five journeys. It runs from almost nothing on a nearly north–south route to 28 per cent on a high-latitude east–west one. Every number is computed from the two distance formulae rather than quoted.Cape Town – London+0.0%3 kmNew York – Madrid+3.0%171 kmSydney – Santiago+12.7%1437 kmLondon – Tokyo+18.2%1737 kmAnchorage – London+28.2%2031 kmextra distance over the shortest routegreat circle vs rhumb linecomputed, not quoted
Fig. 3 Five journeys with the extra distance of the constant-bearing route over the shortest one, computed from the two distance formulae rather than quoted. The range is from nothing to twenty-eight per cent.

The pattern has structure worth naming:

North–south routes cost nothing. A meridian is both a great circle and a rhumb line, so the two coincide. Cape Town to London costs three kilometres out of 9,671 — a rounding error.

East–west routes cost most, and the cost grows with latitude. Anchorage to London is at 60°N and costs 2,031 km, twenty-eight per cent. The reason is that a parallel of latitude is a rhumb line and is not a great circle — following a parallel means constantly turning away from the shortest path.

The equator is the exception. It is a parallel and a great circle at once, so an east–west journey along it costs nothing either. The penalty is a high-latitude phenomenon.

The strange consequence for parallels

That third point deserves its own paragraph, because it is counter-intuitive and it is the crux.

Two cities on the same parallel — New York and Madrid, both near 40°N — are not most quickly reached by travelling due east. The route that holds 40°N the whole way is 5,939 km. The great circle, which arcs north to about 47°N before coming back down, is 5,768 km. The northward detour saves 171 km.

Following a line of latitude feels like the direct route and is not one. Lines of latitude are not straight in any sense the geometry recognises; the only parallel that is a geodesic is the equator.

Measuring the shortest path

Distances here use the haversine formula:

d=2Rarcsin⁡sin⁡2Δφ2+cos⁡φ1cos⁡φ2sin⁡2Δλ2d = 2R\arcsin\sqrt{\sin^2\frac{\Delta\varphi}{2} + \cos\varphi_1\cos\varphi_2\sin^2\frac{\Delta\lambda}{2}}

rather than the spherical law of cosines, which is algebraically equivalent and numerically much worse. The law of cosines takes the arccosine of a quantity very close to one for short distances, and loses most of its significant figures doing so; haversine stays accurate down to metres.

That is a small point of technique and it is the sort of thing that quietly ruins a computation nobody checks. Every distance on this site is asserted against its rhumb counterpart — the great circle must never be longer — and that check would not catch a precision loss but would catch a formula error.

The instrument problem, historically

Why did any of this matter enough to reshape cartography?

Because a ship could hold a bearing and could not hold a great circle. Following a geodesic means continuously adjusting the heading against a curve whose bearing changes every mile, with no way to know the current position beyond dead reckoning. The rhumb line is what could actually be sailed.

So the eighteen per cent was not a choice between a good route and a better one. It was the difference between a route and no route, and Mercator’s chart is the tool that made the sailable route drawable.

The classical method eventually combined both: plot the great circle on a gnomonic chart where it is straight, take waypoints off it, and sail a sequence of rhumb legs between them. Two projections, each used for the single property it has exactly.

The bearing changes, and by how much

A great circle’s compass bearing is not constant, and the amount it changes over a journey is worth quantifying because it is the practical difficulty the whole subject grew out of.

London to Tokyo leaves London on a bearing of about 27° — north-north-east — and arrives at Tokyo on a bearing of about 149°, south-south-east. The heading swings through more than 120° over the route, continuously, with no point at which the ship or aircraft is doing anything but going straight.

That is the counter-intuitive part. Nothing is turning. The meridians are converging underneath, so a straight path crosses each at a different angle, and a compass reading the angle to the local meridian reports a changing number.

London to Tokyo on OrthographicTwo routes. The great circle is 9559 km and is the shortest path on the sphere. The rhumb line holds a single compass bearing the whole way and is 11296 km — 1737 km further, or 18.2 per cent. On Orthographic the rhumb line departs from straight by 2.9e-1 of its own length.LondonTokyosolid: shortest · dashed: constant bearing+18.2% for the rhumbdrawn in Orthographic
Fig. 4 The same route on an orthographic projection, which looks like a globe. The great circle is nearly straight here because the projection is nearly faithful near its centre — and it is still a projection, with its own distortion growing toward the rim.

Where the route goes

A great circle between two points in the northern hemisphere always passes north of the rhumb line, and often surprisingly far north.

The highest latitude a great circle reaches is called its vertex, and for London to Tokyo it is around 70°N — well inside the Arctic Circle, for a route between two cities at 51° and 36°.

This is the geometric content of the observation that polar routes are short. Flights between north-western Europe and east Asia, or between North America and east Asia, cross very high latitudes not as a detour but because that is where the shortest path goes.

It also explains a piece of Cold War history: the shortest routes between the United States and the Soviet Union both went over the Arctic, which is why early-warning radar was built there and why the polar azimuthal projection became the standard strategic map.

Distance on an ellipsoid

Everything here treats the Earth as a sphere, and the real one is flattened by about one part in three hundred.

For distances the difference is around 0.5 per cent — around 30 km on a 6,000 km route, which is negligible for the arguments in this essay and decidedly not for navigation or surveying. The exact computation on an ellipsoid has no closed form and requires either a series expansion or an iterative solution; Vincenty’s method is the standard, and Karney’s 2013 algorithm is the modern one.

The essays here state the sphere assumption because the geodesic structure — that the shortest path is not the constant-bearing path, that it runs poleward, that the excess grows with latitude — is identical on both bodies. The numbers shift; the argument does not.

New York to Madrid, seen four waysThe same two routes on four projections. The gnomonic projection renders every great circle as an exactly straight line, which is what it is for; Mercator renders every rhumb line straight instead. Neither path changed — only the map did.Mercatorgreat circle bows 1.1e-1Gnomonicgreat circle bows not at allEquirectangulargreat circle bows 8.2e-2Orthographicgreat circle bows 1.2e-2solid: shortest · dashed: constant bearingone pair of routes
Fig. 5 New York to Madrid, four projections. Both cities are near 40°N, so the rhumb line runs almost due east and looks like the obvious route. The great circle arcs north of it and is 171 km shorter.

Why the great circle looks like a detour

The perceptual part is worth addressing, because the geometry alone does not explain why the route looks wrong.

A reader looking at a Mercator or equirectangular world map is reading it as though it were a plane. On a plane the shortest path between two points is the straight line, and the eye applies that rule automatically.

The map is not a plane, and the rule does not transfer. What the eye is doing is treating the projection as though it were an isometry, which it cannot be, and the mismatch shows up exactly as the surprise that the curved route is shorter.

Which suggests the useful corrective is not to explain the geometry but to change the map. On a gnomonic chart the shortest route looks shortest, because there it is straight — and the intuition, applied to that map, gives the right answer.

London to Tokyo on GnomonicTwo routes. The great circle is 9559 km and is the shortest path on the sphere. The rhumb line holds a single compass bearing the whole way and is 11296 km — 1737 km further, or 18.2 per cent. On Gnomonic the rhumb line departs from straight by 6.7e+1 of its own length.LondonTokyosolid: shortest · dashed: constant bearing+18.2% for the rhumbdrawn in Gnomonic
Fig. 6 London to Tokyo on a gnomonic chart. Here the eye’s rule works: the straight line is the shortest route, because this projection was built to make that true.

Where the two routes coincide

Three cases, and knowing them is a quick check on any claim about route lengths.

Along a meridian. Every meridian is a great circle and a rhumb line with bearing due north. Cape Town to London is nearly this case, and the excess is three kilometres in 9,671.

Along the equator. The equator is a parallel and a great circle, so an east–west route on it is both.

Over short distances. The two curves diverge as the square of the separation, so at city scale they are indistinguishable and the distinction has no practical content.

Everything else costs something, and the cost grows with latitude and with east–west extent.

Cape Town to London, seen four waysThe same two routes on four projections. The gnomonic projection renders every great circle as an exactly straight line, which is what it is for; Mercator renders every rhumb line straight instead. Neither path changed — only the map did.Mercatorgreat circle bows 1.0e-2Gnomonicgreat circle bows not at allEquirectangulargreat circle bows 2.1e-2Orthographicgreat circle bows 4.6e-3solid: shortest · dashed: constant bearingone pair of routes
Fig. 7 Cape Town to London on four projections. The two routes are nearly coincident in every panel, because the journey is nearly along a meridian — the one geometry where the constant-bearing route is also the shortest.

One last consequence, for anyone reading route maps. An airline’s route map showing curved lines is not decorating them. Those curves are great circles drawn on whatever projection the map uses, and the curvature is the projection’s rather than the route’s — the aircraft flies the straightest path available on a sphere, and the map bends it.

The general lesson is that a map’s straight lines are a property of the map. That sounds obvious stated plainly and is routinely forgotten in practice, because a straight line is exactly the thing the eye reads as meaningful without deciding to. Every projection makes some family of curves straight, and knowing which family is knowing what the map is for.

The result also explains a common confusion about polar routes. They are not chosen to avoid airspace or weather, though those matter at the margin. They are chosen because on a sphere the shortest path between two northern-hemisphere cities on opposite sides of the globe genuinely goes near the pole, and it is the flat map that makes this look like a detour.

The distinction has one more everyday consequence. Distance calculations in software that treats latitude and longitude as plane coordinates — computing a Euclidean distance between two lat-lon pairs — are wrong by an amount that grows with latitude and with separation, and they are wrong in a way that produces plausible small numbers. That is the same error as reading a route off a flat map, made by a program.

What was computed here

Great circles are generated by spherical interpolation between the endpoints — the slerp, in the vocabulary of computer graphics — which stays on the sphere by construction rather than by correction.

Rhumb lines are generated by integrating their defining property, stepping along the sphere holding a constant bearing. That independence is deliberate: the claim tested elsewhere is that Mercator renders these curves straight, and generating them from Mercator’s own ordinate would make the test circular. The integration is verified by requiring each path to arrive at its intended destination, which it does to between 10−1110^{-11} and 10−710^{-7} radians.

The straightness of a projected path is measured as its greatest departure from the chord through its endpoints, as a fraction of that chord. The gnomonic projection renders great circles straight to 3×10−163\times10^{-16} — machine precision — and Mercator renders the same arcs with a bow of about 10−210^{-2}.

What the pictures cannot show

Every figure here is drawn on a projection, so every figure has already made the compromise it is describing. The great circle looks curved in most panels because those projections curve it, and there is no panel showing what it “really” looks like — the page is flat and the sphere is not.

The orthographic panel comes closest, since it looks like a globe, and it is a projection too: it has its own distortion, it shows only a hemisphere, and it flattens dramatically toward the rim.

The figures also cannot show time. A great circle is the shortest distance, and aircraft routes are chosen for shortest time, which brings in jet streams, weather and airspace restrictions. Actual flight paths deviate from great circles for reasons that have nothing to do with geometry.

Who found it, and when

That the geodesics of a sphere are great circles was known to the Greeks; the mathematics of spherical triangles was developed for astronomy long before anyone needed it for navigation.

The practical problem arrived with ocean voyaging. Pedro Nunes, the Portuguese royal cosmographer, established in 1537 that a course of constant bearing is not a great circle and spirals toward the pole — the first clear statement of the distinction, and the reason the rhumb line is sometimes called a Nunes curve.

Great-circle sailing as a routine method dates from the nineteenth century, when accurate chronometers finally made longitude determinable at sea and a ship could know where on the curve it actually was.

Where this goes next

The projection built to make the sailable route drawable is why Mercator exists. The one that straightens the shortest route instead is the gnomonic companion. And the reason no projection can do both is no map is faithful.

Where this goes next

New YorkMadridsolid: shortest · dashed: constant bearing+3.0% for the rhumbdrawn in Mercator Why Mercator exists A ship can hold a compass bearing and cannot easily hold a great circle. Mercator is the answer to one question — what must a map do so that a constant bearing is a straight line — and it answers it exactly. Mercatorgreat circle bows 3.7e-1Gnomonicgreat circle bows not at allEquirectangulargreat circle bows 2.0e-1Orthographicgreat circle bows 9.0e-2solid: shortest · dashed: constant bearingone pair of routes The gnomonic companion One projection turns every great circle into a straight line, and it is the only one that does. It shows less than half the sphere, distorts enormously, and was indispensable for three centuries because of that single exact property. planeK = 0 — unrolls flatcylinderK = 0 — unrolls flatconeK = 0 — unrolls flatsphereK = 1.00torusK = 1.79pseudosphereK = -1.00K computed at each centretwo routes, one answer No map is faithful Not "no map yet", and not "no map at page size". Gaussian curvature can be computed from inside a surface, a sphere has some and a plane has none, and that closes the question permanently.