The aspect is a free choice

Almost every world map aligns its projection with the Earth’s rotation axis. The equator runs across the middle, the poles are at the top and bottom, and the distortion is symmetric about the equator.

There is no mathematical reason for this. The projection does not know about the rotation axis, and pointing it somewhere else is free.

What an aspect is #

A projection is built around an axis. Cylindrical projections have an axis the cylinder is wrapped about; conic ones have the cone’s axis; azimuthal ones have the point the plane touches.

The distortion pattern is symmetric about that axis and fixed relative to it. Rotate the sphere before projecting and the pattern rotates with it.

The three named cases are conventions rather than distinct constructions:

Normal aspect aligns the projection’s axis with the Earth’s rotation axis. For a cylindrical projection this gives the familiar rectangle with straight parallels.

Transverse aspect turns it ninety degrees, so the cylinder wraps pole to pole about an axis through the equator. The line of zero distortion becomes a meridian instead of the equator.

Oblique aspect points it anywhere else.

Nothing about the projection changes. The formulae are the same, the properties are the same — a conformal projection stays conformal in any aspect, because conformality is a local condition and rotating the sphere is locally an isometry.

What changes is where the distortion lands.

Why this is the cheapest improvement available #

The total distortion cannot be reduced, only moved. An aspect change moves it, at no cost and with no loss of any property.

So for any region that is not conveniently placed relative to the equator, choosing an aspect that puts the line of zero distortion through the region is free accuracy. A projection in normal aspect is optimised for the equator, and unless the map is of the equator that is optimisation for the wrong place.

The rule that follows is simple: put the projection’s axis where the map is.

• A region elongated north–south — Chile, Norway, Portugal — wants a transverse cylindrical projection, so the zero line runs along its length.
• A region elongated east–west — Russia, the United States — wants a normal conic with standard parallels inside its latitude range.
• A region elongated along some other direction — Italy, Malaysia, the Great Lakes — wants an oblique aspect, and this is the case most often mishandled.

The one that everyone uses #

The most-used projected coordinate system in the world is a transverse aspect, and most of its users do not think of it that way.

UTM divides the world into sixty zones, each six degrees of longitude wide, and projects each with a transverse Mercator centred on its own middle meridian. Within a zone, the map is conformal and the scale error is small because the zone is narrow.

Every national grid works the same way. The British national grid is a transverse Mercator on a central meridian at 2° west; the same construction appears under different names and datums worldwide.

Which makes transverse Mercator arguably the most consequential projection in use for anything that matters on the ground — surveying, land registry, engineering, military grids — while ordinary Mercator gets all the attention and all the criticism.

The reason the zones are narrow is the reason the aspect was chosen: distortion grows away from the central meridian, so keeping every point within three degrees of one keeps the scale error under about a part in a thousand, which the scale factor of 0.9996 then roughly halves.

Oblique cases worth knowing #

Three, each solving a problem the normal aspect handles badly.

The oblique Mercator puts the line of zero distortion along an arbitrary great circle. It is used for regions elongated on a diagonal — Malaysia, Alaska’s panhandle, Madagascar’s Laborde projection — and for mapping the ground track of a satellite.

The Space Oblique Mercator, designed by John Snyder in 1976 specifically for Landsat imagery, follows the satellite’s ground track continuously as the Earth rotates beneath it. It is conformal along the track and useless for anything else, which is the clearest case in the subject of a projection built to one specification.

Oblique azimuthal projections centred on a city are the standard for any map about distances or directions from one place: air routes from a hub, radio range from a transmitter, the familiar UN emblem centred on the North Pole.

Why the convention persists #

Habit, mostly, and two better reasons.

North is up. Readers expect it, and an oblique map has to explain itself. For a general-purpose map that cost is real.

The graticule reads. In normal aspect, meridians and parallels are simple curves — often straight lines. In oblique aspect they are neither, and the grid becomes a tangle of curves that no longer helps anyone read coordinates off the map.

That second point is the substantive one and it is why oblique aspects are common in technical mapping and rare in atlases. A survey grid does not need a legible graticule; a wall map does.

What the aspect does not change #

Worth stating explicitly, because it is the reason the choice is free.

The projection’s property is unaffected. A conformal projection stays conformal, an equal-area one stays equal-area, and the measured angular deformation and areal error at corresponding points are identical. Rotating the sphere before projecting composes the projection with an isometry, and an isometry changes none of the invariants.

The amount of distortion is unaffected too, in aggregate. The pattern is rigidly rotated, so the histogram of distortion over the whole sphere is unchanged. What changes is which places get which part of it.

That is the whole trade: the same distortion, somewhere else.

The rotation, concretely #

The arithmetic behind an aspect change is worth seeing, because it makes the “free” claim precise.

Rotating the sphere before projecting means computing new coordinates $(\lambda', \varphi')$ relative to a new pole, then feeding those into the ordinary formulae. The transformation is a rotation of the sphere — an isometry — and it is exactly the same operation whichever projection follows.

So an aspect change is a composition: rotate, then project. Since the rotation preserves every distance and angle, it changes none of the invariants, and the projection’s property survives untouched.

That composition structure is why a library can offer arbitrary aspects for any projection without reimplementing anything. This site’s azimuthal projections take a centre for exactly that reason, and the same machinery would extend to the cylindrical family.

A stereographic projection centred between London and Tokyo is still conformal — asserted at the same tolerance as any other conformal projection here — and its distortion is now symmetric about a centre chosen for the map rather than inherited from the Earth’s rotation.

What “north is up” costs #

The convention is worth pricing, since it is the main reason aspects go unused.

For a map of a north–south country — Chile, Norway, Vietnam — a normal-aspect projection wastes most of its sheet and puts the region far from any line of zero distortion. A transverse aspect fixes both at once, and the map is then rotated so north is still up for the reader.

Which is the resolution in practice: choose the aspect for the geometry, then rotate the finished map for the reader. The two decisions are independent, and conflating them is what makes people think north-up requires a normal aspect.

What a national grid actually is #

Assembling the pieces makes the most consequential application concrete.

A national grid is: a datum fixing the ellipsoid and its position; a projection, almost always transverse Mercator; a central meridian chosen to run down the middle of the country; a scale factor slightly below one, converting the tangent construction to a secant one; and false eastings and northings so that no coordinate in the country is negative.

Five choices, of which two are aspect choices — the projection family and where its axis points — and the rest are administrative.

The British national grid uses a central meridian at 2° west, a scale factor of 0.9996012717, and a false origin south-west of the Scilly Isles. None of those numbers is geometric; all of them are decisions made once and now unchangeable, because every map, every land title and every dataset in the country depends on them.

Why sixty zones #

UTM’s zone width has an arithmetic justification worth seeing.

Distortion in a transverse Mercator grows roughly as the square of the distance from the central meridian. At three degrees away — the edge of a six-degree zone — the scale error is about one part in a thousand before the scale factor is applied, and about one part in two thousand after.

One part in two thousand is half a metre per kilometre, which is at the boundary of what ordinary survey work can ignore. Wider zones would exceed it; narrower zones would mean more zones and more boundary-crossing problems.

Sixty zones of six degrees is therefore not arbitrary. It is the width at which the error reaches the tolerance, and the tolerance came from what surveyors of the 1940s could measure.

The gap this essay used to declare, and how it closed #

For a long time the library could only rotate one family. The azimuthal projections take a centre as a parameter, so oblique azimuthal aspects were drawn; the cylindrical and conic families did not, so transverse and oblique cylindrical projections — including transverse Mercator, the one most of the world’s survey data lives in — were described here and not drawn.

Stating the gap was better than quietly drawing the normal aspect and captioning it as the transverse case. It is now closed, and the way it closed is the point of this essay restated as code: one function, applied to every projection in the library, because the operation was never per-projection.

The rotation is done on the three-vector rather than with the spherical-trigonometry identity. Build the unit vector of the point, rotate it so the chosen pole goes to the axis, read the new longitude and latitude off it, and hand those to the ordinary forward map. There are four sign conventions in circulation for the identity and none for the vector route, and the difference between them is a map of a plausibly rotated world.

The claim, now measured #

With the machinery in place the essay’s central assertion stops being an argument and becomes a test.

An oblique projection’s principal scales, areal factor and angular deformation at a point must equal the normal-aspect original’s at the corresponding rotated point. The site checks four cases — Mercator, Mollweide, the Lambert azimuthal equal-area and Gall–Peters, each about a different arbitrary pole — and the worst relative disagreement across all of them is 2.1×10⁻⁸.

That number is not zero, and the reason is worth stating rather than absorbing. The axis-swap check elsewhere on this site comes out at exactly zero because it is an algebraic identity evaluated at one point. This one compares two separate numerical differentiations, at different points, of functions with different conditioning. Mercator is the worst of the four at 8×10⁻⁹ relative, near the pole where its own derivative is diverging; the other three land at 10⁻¹¹.

A rotation built with a wrong sign convention would miss by order one.

The aspect nobody thinks of as one #

Worth ending on the most common aspect choice, which is invisible because it is universal.

Every normal-aspect projection has chosen to align with the Earth’s rotation axis. That is an aspect decision, made by default, and it is the right one only when the map’s subject is arranged latitudinally — which world maps are, and most regional maps are not.

So the question is not whether to use an unusual aspect but whether the usual one was chosen or inherited. For most maps it was inherited, and for a good proportion of them a different choice would have been free and better.

The aspect is also the parameter most likely to be wrong in inherited work, precisely because it is invisible. A projection’s family and property are usually recorded; its aspect is usually the default; and a map of a north-south country on a normal-aspect projection looks unremarkable while wasting most of the available accuracy.

The freedom also extends further than the three named cases suggest. Any rotation of the sphere is a valid aspect, so the aspect is a two-parameter family rather than three options, and the named cases are simply the ones with tidy graticules. A projection centred on an arbitrary point is no harder to compute and is what most regional mapping should be using.

The choice is free and it has a size. Sweeping the axis through every tilt and scoring each one turns the argument into a number.

What was computed here #

The azimuthal projections in this site’s library take their centre as a parameter, so the aspect figures are the same code with a different argument rather than separate implementations.

The claim that the property survives a change of aspect is checked the same way as everything else, and in two forms. The stereographic projection is asserted conformal and the Lambert azimuthal equal-area is asserted equal-area, both in the oblique aspects the figures use, and both pass at the same tolerances as any other projection. And the general form — that every distortion measure at a point equals the normal-aspect original’s at the rotated point — is asserted over four projections about four arbitrary poles, at 2.1×10⁻⁸.

The gnomonic panel additionally asserts its own defining property in the oblique aspect — great circles must still come out straight, and they do to $3\times10^{-16}$ — which is the check that the rotation was applied correctly rather than approximately.

What the pictures cannot show #

The ellipsoidal case. Every oblique figure here is drawn on a sphere, where a rotation is an exact symmetry and the projection is genuinely unchanged. An ellipsoid is symmetric only about its own axis, so an ellipsoidal transverse Mercator is not a rotated ellipsoidal Mercator and has to be derived from scratch. The pictures on this page are therefore correct about the idea and about the wrong body, which is a distinction the caption states and the drawing cannot.

The figures also cannot show the readability cost. An oblique graticule looks like a tangle at a glance, and whether that matters depends on whether anyone needs to read coordinates from the map — which is a question about the user rather than about the geometry.

Who found it, and when #

Lambert constructed the transverse Mercator in 1772, in the same treatise that produced the conformal conic and the cylindrical equal-area. The ellipsoidal version needed for real survey work came much later, from Gauss and then Krüger, which is why the formulae are known as Gauss–Krüger.

UTM was adopted by the US Army in 1947 and became a NATO standard, which is how a projection invented in 1772 came to underlie most of the world’s military and civilian grid systems.

Snyder designed the Space Oblique Mercator in 1976 in response to a problem NASA had and no existing projection solved. He worked it out in his spare time as an amateur, presented it at a conference, and was subsequently hired by the US Geological Survey — which is a good story and also the clearest illustration in the subject that a projection is a specification met.

One thing that story makes concrete is worth keeping. A projection is not a fixed inventory to be chosen from; it is a construction, and the aspect is the part of the construction a user still holds after the family has been picked. Snyder’s case is the extreme version — no existing projection met the specification, so the specification produced a new one — and the ordinary version is the one in this essay, where the specification is met by an existing projection turned to face the right way.

Where this goes next #

The families the aspects apply to are cylinders, cones and planes. The other free parameter within a family is what a standard parallel buys. And for putting both choices to work, which projection is best.