What survives a change of coordinates

A distortion analysis produces six numbers at each point. Two of them are not really about the map.

The two groups #

$h$ and $k$ are the scale factors along the meridian and along the parallel. They are the easiest to compute, the most often quoted, and they depend on which curves have been called meridians and parallels.

$a$ and $b$ are the largest and smallest scale factors over all directions. $ab$ is the areal factor and $\omega$ the angular deformation. None of these mentions a direction, so none can depend on which directions were singled out.

The distinction is exactly the distinction between a matrix’s entries and its eigenvalues. A projection’s derivative at a point is a linear map; $h$ and $k$ are how it acts on two particular vectors; $a$ and $b$ are its singular values, which are properties of the map itself.

Tissot did not have that vocabulary in 1859 and did not need it. The construction — push a circle of directions through and see what comes back — computes the singular values geometrically.

Why it matters in practice #

Three ways this goes wrong, all common.

Quoting $k$ as “the scale factor”. Mercator’s scale factor is often given as $\sec\varphi$, which is $h$ and $k$ both, because Mercator is conformal and they coincide. For a non-conformal projection there is no single scale factor and quoting one means quoting a direction without saying so.

Comparing $h$ across projections with different graticules. An oblique projection’s meridians are not the map’s natural axes, so its $h$ describes something quite different from a normal-aspect projection’s $h$. The principal scales are comparable; the meridian scale is not.

Assuming the maximum stretch is along a meridian or a parallel. It usually is not. For most projections the direction of greatest stretch lies somewhere between, and the whole point of $a$ and $b$ is that they find it without being told where to look.

The relationship #

The two groups are connected by the angle $\theta'$ between the projected meridian and parallel:

$$ab = hk\sin\theta'$$

When the graticule stays orthogonal on the map — as it does for every cylindrical projection in normal aspect — $\theta' = 90°$ and the relation collapses to $ab = hk$. That is why the distinction is easy to miss: for the projections most people meet first, the two groups nearly coincide.

They come apart for oblique aspects, for pseudocylindrical projections away from the central meridian, and for conics — anywhere the graticule is not a right-angled grid.

The leaning is the visible signature. An ellipse whose axes are not along the grid is one whose principal directions are not the coordinate directions, and there $h$ and $k$ are measuring the wrong thing.

The check #

This site asserts the distinction rather than describing it.

The projection’s output axes are swapped — $x$ and $y$ exchanged, which is a genuine change of coordinates that leaves the map the same map — and the six quantities are recomputed. The requirement is that $a$, $b$, the areal factor and $\omega$ are unchanged, while $h$ and $k$ exchange.

The measured drift in the first group is exactly zero. That is the cleanest available demonstration that the first four describe the map and the last two describe the grid, and it runs on every build.

An assertion of this kind is unusual and worth the trouble. Everything else on the site checks that a number is right; this one checks that a number is about the right thing.

There are three tiers, not two #

Swapping the output axes is one change of description. There is a second, and applying it splits the four survivors again.

Multiply the projection’s output by a constant — draw the same map larger. Nothing about the map has changed except the units on the page, and yet $a$ and $b$ both scale by that constant and the areal factor scales by its square. Only $\omega$, which depends on the ratio $(a-b)/(a+b)$, is untouched.

So the six quantities sit in three tiers. $h$ and $k$ depend on how the sphere was parameterised. $a$, $b$ and the areal factor depend on the map’s overall scale, which is a choice of paper size. $\omega$ and the ratio $a/b$ depend on neither.

That is not a defect in the areal factor; it is why “equal-area” is only meaningful once the scale is fixed, and why this site reports areal inflation relative to the equatorial cell rather than in absolute terms. The relative figure is scale-free and comparable; the absolute one is a statement about how large the map was printed.

What the graticule is for #

None of this makes $h$ and $k$ useless. They are how the analysis is done — the four partial derivatives are taken with respect to longitude and latitude, so $h$ and $k$ come out first and everything else is built from them.

They also answer real questions. A surveyor asking how much a map stretches along a north–south line wants $h$, and the coordinate-dependence is not a defect because the question mentioned a coordinate.

The error is only in treating them as descriptions of the projection. They are descriptions of the projection’s behaviour in a chosen direction, which is a different and narrower thing.

The same distinction elsewhere #

It is worth naming the general pattern, because it recurs.

A quantity that changes when the description changes is reporting partly on the description. A quantity that survives is reporting on the object. Physics calls the survivors invariants and organises whole theories around finding them; the same instinct is useful whenever a measurement is being made through a chosen frame.

In this subject the frame is the graticule, which feels so natural that it is easy to forget it was chosen. The prime meridian is at Greenwich by agreement, latitude is measured from the equator by convention, and a projection knows nothing about either.

Why the invariants are the singular values #

Naming the connection makes the distinction less arbitrary, because the same structure turns up wherever a linear map is being described.

A projection’s derivative at a point is a $2\times2$ matrix. Any such matrix can be written as a rotation, then a scaling along two perpendicular axes, then another rotation — the singular value decomposition. The two scalings are the singular values.

$h$ and $k$ are what the matrix does to two particular input vectors, so they change when the input basis changes. $a$ and $b$ are the singular values, which are basis-independent by construction.

That is why the invariance assertion returns exactly zero rather than something small. It is not a numerical coincidence; the quantities are defined in a way that cannot depend on the basis, and swapping the output axes is an orthogonal transformation that singular values are blind to.

What a surveyor uses instead #

A note on where the non-invariant quantities are the right tool, since this essay has been hard on them.

Grid convergence — the angle between grid north and true north at a point — is a coordinate-dependent quantity, and it is one a surveyor needs constantly. So is the scale factor along a specific line being measured. Both are properties of the map together with the grid, and both are exactly what the question asks about.

The error is never in computing $h$ or $k$. It is in reporting one of them as the distortion, which is a claim about the map, when they are answers to a question that mentioned a direction.

An example where it matters #

A concrete case, since the distinction can seem academic.

Suppose two projections are being compared for a country and one is quoted as having a meridian scale factor of 1.02 while the other has 1.01. That looks decisive and says nothing, because the two projections may put their maximum stretch in completely different directions.

The first might have $h = 1.02$, $k = 1.02$ — conformal, with a uniform two per cent scale error. The second might have $h = 1.01$, $k = 0.99$ — a one per cent stretch one way and a one per cent squeeze the other, which is a shape distortion the first does not have at all.

Comparing $h$ ranked them wrongly. Comparing $a$ and $b$, or $\omega$ and the areal factor, would have separated them immediately.

What the graticule is, historically #

Worth remembering that the coordinate system whose influence this essay is trying to isolate was itself a choice, and a fairly recent one.

Latitude from the equator is natural enough — the equator is defined by the rotation. Longitude from Greenwich is not: it was settled by international conference in 1884, and Paris, Cadiz, Washington and the Canary Islands all had serious claims and periods of use.

So the prime meridian is a diplomatic outcome, and every $h$ and $k$ computed on this site is measured relative to a grid anchored by an 1884 vote. That the invariants do not care is the point of them.

Where else the distinction appears on this site #

The same separation recurs across the collection, and noticing the pattern makes it easier to apply.

The distortion measures split into invariants and coordinate-dependent quantities. The families split into what a projection is built from and what it preserves. The aspect is a change of coordinates that leaves every property alone.

In each case the useful move is the same: ask what would change if the description changed, and treat whatever survives as the content. That is not a deep principle so much as a habit, and it is one this subject rewards unusually well because the descriptions are so conventional — a graticule anchored at Greenwich, a projection aligned with the rotation axis, a taxonomy inherited from an era of geometric construction.

Measuring rather than naming is the same habit applied to the projection’s own description of itself.

One practical closing note. When reading somebody else’s distortion analysis, the first thing to check is which quantities are quoted. A table of meridian and parallel scale factors is a table about that author’s graticule; a table of principal scales, areal factors and angular deformations is a table about the maps. Both are useful and only the second is comparable across sources.

The habit generalises past this subject, which is why it is worth the essay. Anywhere a measurement is made through a chosen frame — a coordinate system, a unit, a baseline, a reference category — the same question applies: what would change if the frame changed. Whatever survives is the finding, and whatever does not is partly a description of the apparatus.

The distinction is also what makes a projection library’s output comparable at all. Two implementations may parameterise the sphere differently, order their outputs differently, or scale their results differently, and their principal scales, areal factors and angular deformations will still agree. That is why this site’s audit can compare twenty-one projections written by different hands over four centuries.

The check this site runs is unusual enough to be worth restating as a recommendation. Most tests verify that a computed number is correct. This one verifies that a number is about what it is claimed to be about, by changing something that should not matter and requiring the answer not to move. Where a quantity is supposed to be intrinsic, that test is available and is rarely written.

The essay’s practical residue is one habit: when a distortion figure is quoted, ask which quantity it is. If it is a meridian or parallel scale factor, it is a statement about the map and the grid together and is not comparable with anything computed on a different graticule. If it is a principal scale, an areal factor or an angular deformation, it is a statement about the map and can be compared with anything.

There is a last reason the distinction is worth the trouble on this particular site. The whole method here is to measure a projection’s claimed property rather than accept it, and a measurement made in the wrong quantity would defeat that entirely — a conformality test computed from h and k alone would be testing something that depends on where the prime meridian was put. The invariants are what make the audit an audit.

The principal scale factors survive a change of coordinates and so does the direction they belong to. It is the third invariant, and it is the one nobody prints.

What was computed here #

All six quantities come from the same four partial derivatives, obtained by Richardson-extrapolated central differences.

The route follows Tissot: $h$ and $k$ from the lengths of the two partial-derivative vectors divided by the corresponding distance on the sphere; the areal factor from the Jacobian determinant divided by the area element; $\sin\theta'$ from the ratio of those; and $a$, $b$ from the half-sum and half-difference of $\sqrt{h^2+k^2+2hk\sin\theta'}$ and $\sqrt{h^2+k^2-2hk\sin\theta'}$.

The relation $ab = hk\sin\theta'$ holds identically by construction, which makes it a useful hand-check when adding a projection and a poor test of the implementation — it would hold even if the derivatives were wrong. The invariance assertion is the real test, and it is independent of it.

What the pictures cannot show #

An invariant is defined by what it does not do, and a figure showing that a number did not change under a transformation is a figure of two identical numbers. The check is real and it is not illustratable, which is why this essay states the measured drift rather than drawing it.

The leaning ellipses in the sinusoidal figure come closest: they show a case where the coordinate directions are visibly not the principal directions, which is when the distinction has visible consequences.

A field whose directions are the map’s #

Take the velocity of the ground under a coordinate, drawn as arrows on a map. The speed at each point is a property of the plate; the bearing shown on the paper is a property of the plate and the projection together, which is why any such figure has to name the projection it is drawn in.

A vector field on a sphere makes the invariance question concrete in a way a scalar cannot. Two maps of the same plate motion show arrows pointing measurably differently and neither is wrong — the same statement as the one about h and k above, one dimension up. What survives is the speed and the angle between two vectors at a point on a conformal map; what does not is the direction drawn on the sheet.

Who found it, and when #

Tissot’s construction is from 1859, extended in 1881. The singular value decomposition, which is what the construction computes, was developed independently by Beltrami and Jordan in the 1870s — near-simultaneously and without either connecting it to cartography.

The general principle that the meaningful quantities are the ones surviving a change of description is older and more diffuse, and it is what Klein’s Erlangen programme of 1872 made explicit for geometry: a geometry is the study of the properties preserved by a group of transformations.

The projection case is a small instance of a large idea, and it is one where the temptation to quote the non-invariant quantity is unusually strong, because $h$ and $k$ are the ones that fall out of the arithmetic first.

The ordering is the trap, and it is worth stating as advice. The quantities that come out of the arithmetic first are $h$ and $k$, they are easy to compute, they have obvious meanings, and they are the ones a reader will find quoted — and they are exactly the pair that depends on the coordinate system somebody happened to use. The invariants take one more step and are the only ones that describe the map. Nothing in the arithmetic marks the difference, so the discipline has to come from knowing which group the description is supposed to be indifferent to.

Where this goes next #

The construction all six quantities come from is Tissot’s indicatrix. The two invariants that matter most are the two ways a map is wrong. And for what happens when the invariants are actually measured rather than assumed, measuring instead of naming.