The line drawn straight on the page is a route

Eleven essays in this field take a route defined on the sphere — shortest, constant-bearing, quickest, altitude-constrained — and ask what the map does to it. Nobody navigating with a chart does that.

They lay a straight edge between two points on the paper and follow the line — not the shortest route, not the great-circle vertex, not anything chosen on the sphere at all. The ground curve that results is a real route, with a real length, and it is a property of the projection rather than of the sphere. It has a name on exactly two members of this library.

The two named cases #

Mercator. The straight line on the page is the rhumb, the constant-bearing track, and this is the entire reason the projection exists. Why Mercator exists derives it: the projection is constructed so that a curve crossing every meridian at a constant angle is a straight line, because that is the curve a ship steering one compass bearing follows.

Measured here, the ground length of the page-straight London–Tokyo line on Mercator is 11,296 km, and the rhumb distance between the same two places is 11,296 km — agreeing to 1.3 parts per million, which is the sampler’s own noise.

The gnomonic. The straight line is the great circle, and that is the entire reason a gnomonic chart is carried alongside a Mercator one. The gnomonic maps every great circle to a straight line, so the shortest route can be drawn with a ruler and its waypoints transferred to the Mercator sheet for steering.

Measured, 9,559 km against a great-circle distance of 9,558 — agreeing to four decimal places of a per cent.

Everything else is an unnamed curve #

The other eight projections in the audit produce a track that is neither. Its length is between 17.1 and 19.6 per cent above the shortest route, and it strays up to 3,075 km from it.

That is not a small quantity. Nineteen per cent on a transpacific route is over two thousand kilometres, which on any real journey is days and, for an aircraft, more fuel than the aircraft carries.

The rhumb is not the disaster, and that is worth stating #

Mercator’s 18.2 per cent looks bad in the table and is the projection’s defence rather than its indictment.

A rhumb is a route somebody can actually steer, and the route that is quickest is a third thing again. A constant compass bearing needs no navigation at all beyond a compass, no recomputation, no waypoints and no position fixes — which for four centuries was the difference between a route a ship could follow and one it could not.

Eighteen per cent is what that costs on a long high-latitude route, and it was worth paying. The shortest route is not straight sets the comparison up; this rung supplies the number, and the number is the price of a navigational method rather than a defect of a map.

Length is the right instrument, and comparing tracks is not #

The identification of the two named cases is made by length rather than by comparing the drawn track with the named curve point by point, and the reason is a measurement error worth recording.

Comparing two sampled curves means comparing two lists of points, and the two samplers parameterise their curves differently: one walks linearly along the page, the other by the rhumb’s own construction. Matching them by index therefore measures the parameterisation. Doing so returned 376 kilometres of disagreement between two curves that coincide.

Comparing them as curves — nearest point on one to each point of the other — improved it to 19 kilometres, which is half the sample spacing and is a measurement of the sampler.

Length is exact. Two curves with the same endpoints and the same length, one of which is known to be a rhumb, are the same curve; and the comparison is a scalar with no parameterisation in it. The claim this rung makes is a claim about lengths, and it is made at one part in a million rather than at one part in six hundred.

The azimuthal family, and a familiar shape #

Centre an azimuthal projection on the midpoint of the route, and every member of the family draws the great circle straight — stereographic, orthographic, equidistant, Lambert azimuthal and gnomonic alike, to fourteen decimal places.

The reason is immediate once stated. A route through the centre of an azimuthal map is a radial line, and every azimuthal projection preserves the bearing from its own centre by construction. So the property belongs to the family and not to the member.

This is exactly the shape of the reach ladder’s first finding, where every azimuthal projection draws a range ring as a circle and only the equidistant member spaces the rings correctly. A family property is mistaken for a member’s, twice, on two different questions, by the same reasoning.

What the gnomonic alone has is that it draws every great circle straight, including those nowhere near its centre. Centred well off the route, it still returns the geodesic to four decimal places of a per cent while the other four are out by between 0.37 and 1.16.

Why the gnomonic is the only one #

The gnomonic is the projection from the centre of the sphere onto a tangent plane. A great circle is the intersection of the sphere with a plane through the centre; projecting that intersection from the centre onto another plane gives the intersection of two planes, which is a line.

So the property is not an approximation or a design choice; it is what central projection does to plane sections, and no other construction has it.

It comes at a price the family’s other members do not pay: the gnomonic shows less than a hemisphere, its distortion grows without bound towards the rim, and the ratio between the largest and smallest scale on any usable sheet is enormous. A chart that gets every route right and every shape wrong is exactly the specialised instrument a navigator carries alongside a general-purpose one.

What the drawn line actually is elsewhere #

On a projection that is neither, the page-straight line pulls back to a curve with no standing name. It is the image of a straight line under the inverse projection, which is a perfectly well-defined object and has no navigational meaning.

It is not a curve of constant bearing. It is not a shortest path. It is not a curve of constant anything — the bearing along it varies, the curvature varies, and there is no quantity it holds fixed.

That is worth saying because a reader of a map may reasonably assume that a straight line means something. On two projections it does, and both of those projections were designed so that it would.

Reading the numbers back to front #

The audit can also be read as a statement about what a projection is for, which is this field’s standing position.

A projection on which the drawn line is a useful route is a navigational instrument: the ruler does part of the work, and the map has been designed to make that true. Mercator and the gnomonic are the two, and both were designed for it.

A projection on which the drawn line is meaningless is a general-purpose map, and it has traded that particular convenience for something else — area, shape, or a compromise between them. Nothing is wrong with that, and a reader who lays a ruler on one is using an instrument for a purpose it does not have.

The two categories are not a matter of degree. Between the gnomonic at 0.00 per cent and the next-best member at 17.1 there is nothing at all.

The inverse is where the measurement lives #

Computing this needs the inverse projection at every point along the segment, and for most of the library the inverse is Newton’s method on the forward map.

That is where the measurement nearly went wrong. The gnomonic has two basins — a point and its antipode both project to positions on the page — and an initial guess interpolated in longitude and latitude lands in the wrong one part-way along a long route. The first version of this measurement returned a track 2,146 per cent too long, which is not a plausible route and is a solver that converged to the far side of the world.

The fix is continuation: walk the inverse along the segment, using each solved point as the guess for the next. That keeps the solver in the basin it started in, and it is the standard remedy whenever an inverse has more than one branch — the same problem a retroazimuthal projection has everywhere, arriving as a numerical difficulty rather than as a property of the map.

The eight are all near the rhumb, and that is a coincidence #

There is a pattern in the numbers that is worth flagging as a coincidence rather than as a result.

The eight unnamed cases cost between 17.1 and 19.6 per cent, and the rhumb costs 18.2. So they cluster around the rhumb, within a couple of points either side.

That is not because they are approximating a rhumb. It is because the page-straight line on almost any projection of a long east–west route runs roughly along a parallel — the projections in question all draw parallels as horizontal lines or nearly so — and a route along a parallel is very nearly a rhumb.

Change the route to one running north–south and the cluster disperses, because a meridian is both a rhumb and a great circle and every projection draws it straight. The clustering is a property of this pair of endpoints, and the essay’s own figure would look quite different for London to Cape Town.

Where the model stops #

Everything here is on a sphere. On the ellipsoid the great circle is replaced by a geodesic and the rhumb by an ellipsoidal loxodrome, and the gnomonic’s exact property is lost — an ellipsoidal geodesic is not a plane section, so central projection does not straighten it. The gnomonic remains an excellent approximation and stops being exact, which is a rung this ladder has not written.

Nor is anything here about a route that is drawn in pieces. A navigator laying a ruler between waypoints rather than between endpoints gets a piecewise-straight page track, whose ground length falls towards the geodesic as the pieces get shorter — which is flying a curve in straight legs from the other end.

What a reader is entitled to assume #

The practical conclusion is a short list, and it is a list about maps rather than about routes.

On a Mercator sheet, a straight line is a course to steer, and following it takes 18 per cent further on a long high-latitude route than the shortest path does. That is the deal the projection offers and it is stated on the sheet by four centuries of convention.

On a gnomonic sheet, a straight line is the shortest route, and the sheet is unusable for almost everything else.

On an azimuthal sheet centred on one of the two places, a straight line to that place is the shortest route. That is why an azimuthal equidistant map centred on a transmitter is the right map for a radio path and why the same map is the wrong one for any other route on it.

On everything else, a straight line is a curve with no properties, and the ruler is measuring the paper.

The last case is the common one and it is the one no map says anything about. A general-purpose atlas sheet invites a ruler exactly as much as a chart does, and answers it with a number that means nothing.

Who found it, and when #

Mercator’s construction is from 1569 and the reason for it was stated on the map itself. The gnomonic’s great-circle property is much older — it is in Thales by attribution and certainly in the classical treatments of the analemma — and its pairing with Mercator as a navigational method dates from the nineteenth century, when great-circle sailing became routine.

What has not been done, as far as this collection can tell, is the audit: taking the page-straight line seriously as a route on every projection and measuring what it costs. The reason is probably that it is only an interesting question if somebody might actually do it, and for most of the history of navigation the two charts that make it meaningful were the two charts on the table.

The measurement as a test of the library #

There is a secondary use for this audit that is worth recording, because it is how the site’s own gate uses it.

The claim Mercator straightens the rhumb and the claim the gnomonic straightens the great circle are both checkable to a part in a million by this machinery, and both are properties of the projections’ formulae rather than of anything approximate. So the audit is a test of the library: if a change to the projection code broke Mercator’s construction, this measurement would catch it immediately and no distortion measure would.

The refusal half is equally sharp. Every projection that is not one of the two must fail both tests, and all eight do — none is within a hundredth of a per cent of either named length. A version of the library in which everything straightened everything would be a library with a bug in its inverse, and this check would find it.

That is why the assertion is written as two positive claims and eight negative ones rather than as a tolerance on a table.

The affordance belongs to the medium, not to the map #

One reading of the audit is worth separating from the arithmetic, because it explains why the question keeps being asked in the first place.

A map does not offer straight lines. A ruler does. Nothing about a projection invites a reader to join two points with a straight segment; what invites it is the straight-edge on the desk, the line tool in the drawing program, the two-finger gesture that measures a distance on a screen. The affordance is a property of the instrument being laid on the sheet, and it is identical on every sheet.

Which is why the answer varies and the expectation does not. The reader’s gesture is the same on a Mercator, a gnomonic and a Robinson, and the curve on the ground is different in each case, unnamed in eight of the ten. There is no feedback: the line looks the same, the map accepts it, and the ground never reports back.

And it explains the two exceptions’ status. Mercator and the gnomonic are the two projections built so that a particular instrument gives a particular answer, and both were designed in an era when the instrument was the point — a chart existed to be worked on with a straight-edge and dividers. The named cases are the projections that took the affordance seriously; the rest simply inherited it.

A modern screen has quietly widened the problem. Every mapping interface offers a line tool, every one of them draws a straight segment in page coordinates, and the underlying sheet is a Web Mercator on which that segment is a rhumb line — so the world’s most-used map has an affordance whose answer nobody chose and almost nobody knows.

Where the ladder goes next #

Nine rungs of this anchor treat a route as a curve chosen on the ground and drawn on a map. This one reverses the order — chosen on the map and realised on the ground. What neither direction has asked is what happens when the route must satisfy a constraint that is itself drawn on the page, such as staying within a corridor or clear of a boundary.