The gnomonic companion

There is exactly one projection on which every shortest route is a straight line. It shows less than a hemisphere, its distortion is unbounded, and it was worth using anyway.

The construction #

Put a plane tangent to the sphere at some point. Put a light at the centre of the sphere. Project every point of the surface outward through the plane.

That is the gnomonic projection, and it is one of the few in common use that really is a geometric projection rather than a formula named after a shape.

The reason great circles come out straight follows in one line. A great circle is the intersection of the sphere with a plane through the centre. Rays from the centre through that circle all lie in that plane. A plane intersected with the image plane is a straight line. So every great circle maps to a straight line, and the converse holds too.

No other projection can do this in general, because the property is essentially the definition of the construction — the light has to be at the centre for the great-circle planes to pass through it.

What it costs #

Everything else.

Only points on the near side are visible at all, since a point on the far hemisphere projects backward through the light. The equator of the tangent point maps to infinity, so the useful area is strictly less than half the sphere and in practice much less — beyond about 70° from the centre the distortion is unusable.

The site measures it at 89.7° of maximum angular deformation and an areal factor reaching nearly 200 over the region it draws, and both grow without bound as the horizon is approached.

It is not conformal and not equal-area. It preserves one thing exactly and nothing else at all, which makes it a good illustration of a general point: a projection is a specification met, not a compromise struck.

The composite method #

The two exact properties — Mercator’s straight rhumbs and the gnomonic’s straight great circles — were used together, and the method is a neat piece of engineering.

Plot the route on a gnomonic chart. Draw a straight line between the endpoints; that line is the great circle. Read off a series of points along it, conventionally where it crosses convenient meridians.

Transfer those points to a Mercator chart. Join them with straight segments. Each segment is now a rhumb line with a single constant bearing, which the ship can actually steer.

The result is a polyline approximating the great circle, made of legs each of which is sailable. More waypoints means a closer approximation and more course changes, and the navigator chooses the trade.

Two projections, neither of which can do the other’s job, composed into a method that does both. This was standard practice for ocean passages into the twentieth century.

How many legs it takes #

The trade the navigator makes has a rate, and it is worth computing rather than describing, because it explains why the method was practical rather than merely correct.

Take London to Tokyo. The great circle is 9,559 km and the single rhumb line is 11,296 km — an excess of 1,737 km, or 18.2%. Now break the great circle into equal legs and sail each as a rhumb:

• 2 legs: 10,071 km, excess 513 km (5.4%)
• 4 legs: 9,710 km, excess 151 km (1.6%)
• 8 legs: 9,597 km, excess 39 km (0.40%)
• 16 legs: 9,569 km, excess 10 km (0.10%)

The excess falls by a factor of about four each time the leg count doubles, which is the $1/n^2$ behaviour a chord approximation to a smooth curve gives. Eight course changes on a route of nine and a half thousand kilometres recover all but four hundred metres in a thousand.

That is why the method survived three centuries. The first few waypoints do nearly all the work, and a navigator willing to alter course eight times over a two-week passage was giving up essentially nothing — which is a far better bargain than the raw eighteen per cent suggests.

Where it is still used #

The projection has outlived the navigational method.

Seismology. Waves travel great-circle paths through the Earth, so a gnomonic chart centred on an epicentre shows their routes as straight lines.

Radio propagation. Long-distance signals follow great circles, and antenna bearings are computed on them.

Aviation planning. Great-circle routes remain the basis for flight planning, and the gnomonic projection is still the natural chart for laying them out even when the arithmetic is done numerically.

Panorama and cube-map rendering. Each face of a cube map is a gnomonic projection of the sphere onto a square, which is why straight lines in a rendered scene stay straight within a face.

That last one is worth noticing: the projection is now used far more in computer graphics than in navigation, for the same reason it was used at sea. Straight things stay straight.

The name #

Gnomon is the part of a sundial that casts the shadow, and the projection is named for the analogy: a point at the centre casting shadows of the sphere’s features onto a plane.

It is the oldest projection known — Thales is traditionally credited with it around 580 BC, which would make it substantially older than the idea that the Earth is a sphere was uncontroversial. It was used for star charts and for sundial construction long before anyone needed it for maps of the Earth.

That figure carries a nice check on the whole idea. Every meridian is straight and every parallel except the equator is curved, and the rule separating them is exactly whether the curve is a great circle.

The horizon, and what happens near it #

The projection’s limitation is worth quantifying rather than merely noting, because it decides how the projection is used.

A point at angular distance $c$ from the centre maps to distance $\tan c$ from the origin. At 45° that is 1; at 60° it is 1.73; at 80° it is 5.7; at 89° it is 57. The growth is not gradual near the end — it is a tangent, and it diverges.

Which means a gnomonic chart covering more than about 60° from its centre is dominated by the outer regions, and one covering 80° is unusable. In practice charts were made for a specific ocean passage, centred near its midpoint, and covered whatever the route needed.

That is why gnomonic charts were published as a series rather than as a world map: a chart for the North Atlantic, one for the North Pacific, one for the Southern Ocean. Each is centred where its routes are.

Why no other projection can do it #

The uniqueness deserves a sentence, since “the only projection that does this” is a strong claim.

Suppose a projection maps every great circle to a straight line. Great circles through a given point form a pencil; their images are lines through the image point. Taking three great circles through two points and applying the same argument at both constrains the map heavily, and the classical result — a consequence of Beltrami’s theorem — is that on a surface of constant curvature the only such maps are the gnomonic ones, up to a projective transformation of the plane.

So it is not that nobody has found another. There is not another.

What it does to the graticule #

Reading the gnomonic graticule is a good exercise in the projection’s single rule, because everything about it follows from one criterion.

Meridians are straight, because every meridian is a great circle.

The equator is straight, for the same reason.

Every other parallel is curved, because a parallel other than the equator is a small circle rather than a great one.

That is the whole of it, and it is a useful check: any curve on a gnomonic chart is straight if and only if it is a great circle on the sphere. The projection is a great-circle detector.

The other azimuthals, for comparison #

The gnomonic is one of a family, and the family’s members differ only in where the light is placed.

Light at the centre gives the gnomonic: great circles straight, less than a hemisphere shown, unbounded distortion.

Light at the antipode gives the stereographic: conformal, the whole sphere except one point, circles mapping to circles.

Light at infinity gives the orthographic: the view from far away, a hemisphere, looking like a globe.

Three projections, one construction, one parameter — and their properties are completely different. Which is a compact illustration that a construction does not determine a property.

Five azimuthal projections share the radial graticule and the property of preserving directions from the centre, and differ in everything else: one is conformal, one equal-area, one straightens great circles, one preserves distances from the centre. Membership of the family predicts none of that.

Why it never became a general-purpose map #

The projection is nearly three thousand years old and has never been used for a world map, which is worth a sentence.

It cannot be. Less than a hemisphere is representable at all, and the usable region is much smaller. There is no aspect, no parameter and no variant that fixes this — the divergence at 90° from the centre is structural, and it follows from the light being at the sphere’s centre, which is the same fact that gives the projection its one property.

So the gnomonic is the clearest case in the subject of a projection that is exactly right for one job and unusable for everything else. Naming the purpose before the property is not a general principle it illustrates so much as a principle it embodies completely.

The projection also has a quiet second life in mathematics, where it is the standard model for showing that the sphere’s geodesics behave like lines. Beltrami used exactly this map in 1868 to build the first model of hyperbolic geometry, transferring the gnomonic idea to a surface of constant negative curvature — which makes a navigational chart from antiquity part of the machinery that settled whether Euclid’s parallel postulate was independent.

It is worth ending on the projection’s economy. One property, exactly; everything else abandoned; less than half the sphere shown. By any general measure it is the worst projection in the library, and for the one question it answers there is nothing else. That combination — extreme specialisation, complete uselessness elsewhere — is the clearest possible illustration that a projection is not good or bad but fit or unfit.

The construction also makes it the natural projection for anything defined by planes through the centre of a sphere. Crystallography’s stereographic and gnomonic projections of crystal faces work this way, as does the representation of orientations in structural geology — the sphere is not the Earth in either case, and the property being exploited is the same one.

The projection is also a good test of whether a claim about maps has been checked. “Great circles are straight lines on the gnomonic projection” is repeated everywhere and is true, and it is the sort of statement that could equally have been repeated for centuries without being true — which is why this site measures it at 3×10⁻¹⁶ rather than repeating it too.

There is one more reason the projection survives in graphics. A cube map’s six faces are gnomonic projections onto squares, chosen because straight lines in the scene stay straight within a face — the same property, exploited for the same reason, three thousand years later and for a sphere that is not the Earth.

The projection is finally a useful corrective to the idea that a good map is one with little distortion. By any aggregate measure this is the worst projection in the library — 89.7° of angular deformation, an areal factor approaching 200, less than half the sphere shown — and for the question it answers there has never been an alternative in three thousand years.

Which is the shape of nearly every argument on this site. A projection is not good or bad; it is fit or unfit for a stated job, and the gnomonic is the case where the job is narrowest and the fit is exact.

It is also the projection that most cleanly separates the two questions this whole collection keeps returning to. What does this map preserve, and what is it for. For the gnomonic both have single-word answers, and the answers are the same word — which is as close to a designed object as this subject gets.

The gnomonic gives the route and cannot give the instructions. What lies between the two is the plan a navigator files.

What was computed here #

The straightness claim is measured rather than asserted. A great-circle path is generated by spherical interpolation between endpoints — which stays on the sphere by construction — then projected, and its greatest departure from the chord through its endpoints is measured as a fraction of that chord.

On the gnomonic projection that departure is $3\times10^{-16}$. On Mercator, for the same arcs, it is about $10^{-2}$ — four thousand times larger, and visible as the bow in the figures.

The test is required to discriminate. If Mercator also rendered great circles nearly straight, the gnomonic claim would be vacuous, so the gate asserts that the same arcs bow measurably on Mercator. It also runs the mirror-image check for Mercator’s own claim about rhumb lines.

The projection’s own domain is handled explicitly: points more than about 80° from the centre are excluded from the figures, because the projection sends them to infinity and a naive plot would produce coordinates in the millions.

What the pictures cannot show #

Most of the sphere. The gnomonic panel in the hero figure shows the region around the route and nothing else, and a full-world gnomonic map does not exist — the far hemisphere has no image and the near horizon is infinitely far away.

The composite navigational method also cannot really be drawn on a single page, because it needs two charts side by side and the transfer between them is a manual operation. The figures show each projection doing its own job; the method is the sentence connecting them.

The control it provides for a stored segment #

The projection’s defining property has a modern use this essay’s period could not have anticipated: it is the control that makes a measurement of other projections meaningful.

A straight segment between two stored points departs from the ground it claims by hundreds of kilometres on a cylindrical projection. On the gnomonic the same measurement returns 2.3 × 10⁻⁷ metres — a fifth of a micron, which is the sampling and the arithmetic and nothing else. So the departure is a property of the plane rather than of straightness, and there exists a plane in which it vanishes exactly.

It also supplies a control nobody designed it for. The procedure that finds 718 kilometres of departure between a stored straight segment and its own route on a web map finds a fifth of a micron here, because on this projection a great circle is a straight line — and the vertices a stated tolerance buys on a plane where it is not follow the square root of the quadratic law: a hundredfold tighter line costs sixteen times the vertices.

Who found it, and when #

Attributed to Thales, around 580 BC, which makes it the oldest projection in continuous use. Its early applications were astronomical — projecting the celestial sphere onto a plane for star charts — and horological, for laying out sundial faces.

Great-circle sailing as a routine practice is much later. It required a reliable way to know one’s position, which meant the marine chronometer and the determination of longitude, and so belongs to the late eighteenth and nineteenth centuries. The gnomonic chart was available for two thousand years before the problem it solved became a practical one.

The composite gnomonic-plus-Mercator method was standard training for merchant and naval officers until satellite navigation made continuous course correction trivial, at which point ships began steering great circles directly and the method quietly retired.

What survives the retirement is the projection’s one exact property, which is why it is still the right tool whenever a great circle has to be drawn straight rather than steered.

Where this goes next #

The problem it solves is the shortest route is not straight. Its partner in the classical method is why Mercator exists. And for where it sits in the usual taxonomy, cylinders, cones and planes.