Tissot stops at the first derivative

A conformal map has no angular deformation. That is the whole meaning of the word, it is checked here at several hundred points before the word is printed, and on the Mercator projection at 20° east, 40° north the measured value is zero to the last figure the arithmetic carries.

At the same point, a great circle passing through on a bearing of 70° is drawn as a curve that turns through 0.839 radians for every radian of arc it covers. Nothing about that number is available from the indicatrix, from the principal scale factors, from the areal factor or from ω. They are all first-order quantities and this one is not.

What a first derivative can and cannot hold #

The projection is a function from the sphere to the plane. Its derivative at a point is a two-by-two matrix — the Jacobian — and everything this site has measured until now is a number read off that matrix.

The two principal scale factors are its singular values. The areal factor is its determinant. The maximum angular deformation is a function of the ratio of the singular values. Tissot’s indicatrix is the matrix, drawn: the image of an infinitesimal circle under a linear map is an ellipse, and the ellipse carries exactly the information the matrix carries and no more.

A linear map takes straight lines to straight lines. So at the level of the first derivative, every projection is perfect in one specific sense: a curve through the point is taken to a curve through the image with the correct tangent direction, up to whatever the indicatrix’s ellipse does to angles. Bending is not in the picture at all, because bending is what the second derivative describes and the first derivative has been asked for.

That is not a defect in Tissot’s construction. It is a statement of what the construction is for, and the indicatrix is a limit already measured the price of forgetting the word infinitesimal in it. This essay is about a different omission: not how far the first-order description reaches, but what it never described at all.

Two quantities, and they are all there is #

Take a geodesic through the point, parameterised by arc length $s$ on the sphere, and let $Q(s)$ be its image in the map. The tangent $Q'$ is first-order information. The second derivative $Q''$ is the whole of the second-order information in that direction, and it has exactly two components:

• the part across the tangent, which turns the image;
• the part along the tangent, which changes the image’s speed.

Normalising each by the local speed so that neither changes if the whole map is enlarged gives the pair this ladder is about:

$$\text{flexion} = \frac{Q''\cdot\hat n}{|Q'|}, \qquad \text{skewness} = \frac{Q''\cdot\hat t}{|Q'|}$$

Flexion is radians of turning in the map per radian of arc on the sphere. Skewness is the rate of change of the logarithm of the local scale along the same direction, per radian of arc. Both are zero for an isometry. No projection of the sphere is an isometry — that is the theorem no map is faithful derives from the curvature — so no projection has both zero everywhere.

The names are Goldberg and Gott’s, who introduced the pair in 2007 to rank world maps by something Tissot’s apparatus cannot express. Every number printed here is differenced from this site’s own forward maps rather than quoted from theirs.

Why the first-order picture cannot be patched to include it #

A reader who has met the indicatrix might reasonably expect the bending to be recoverable from it. Scale varies from place to place; a varying scale ought to bend things; so surely the ellipse plus its neighbours contains the answer.

Half of that is right and the half that is right is the interesting half. The bending is recoverable from how the first-order quantities vary across the map — flexion is a derivative of the Jacobian, and the Jacobian is what the indicatrix draws. What is not true is that the indicatrix at a point contains it. A quantity computed from the rate of change of a field is not a property of the field at one place, and the site’s own habit of drawing a scattering of ellipses across a world map is precisely the presentation that hides it: the ellipses are drawn far enough apart that the eye cannot difference them.

So the second-order quantities are not a competitor to Tissot’s. They are the next term, they are computable from the same forward map, and this site had never computed one.

The measurement, and the step it is made at #

Three evaluations of the forward map along the geodesic, at $-\delta$, $0$ and $+\delta$, give the first and second derivatives by central differences. The step is a decision with two failure modes pulling opposite ways: the truncation error of the difference falls as $\delta^2$ and the cancellation error of subtracting nearly equal coordinates grows as $1/\delta^2$.

Ten to the minus three radians — about six kilometres on the Earth — sits in the valley between them, and the code asserts that it does by measuring both slopes rather than by looking at a number and finding it plausible:

The two decades either side of the chosen one are noisier than the one between them, which is what a valley looks like and what an assertion can be written about.

The geodesic itself is exact rather than integrated. A great circle is the rotation of a point about the pole of the plane containing the point and the direction, so a point at arc length $s$ is one line of trigonometry. Integrating a path and then differentiating it twice would put the integrator’s truncation error into the answer, in the one place it could not be tolerated.

The bow of a drawn line, which is the same number by another route #

Flexion is defined as a limit at a point, and a limit is a hard thing to trust on its own. The quantity a reader actually meets is the bow: a great circle drawn on a map departs from the straight line ruled between its ends, and the departure is measurable with no derivatives in it at all.

For an arc that turns at a constant rate the departure has a closed form. The sagitta of a circular arc is $\ell^2/8R$, so the departure divided by the chord is the total turning over eight — and the total turning is the flexion times the arc length in radians. The departure is therefore quadratic in the length and the relative departure is linear in it.

Both exponents come out where the algebra says: 1.997 and 0.997 across seven lengths spanning six doublings. And the coefficient of the linear law is the flexion, which is the check worth having:

Eight times the intercept of that fit is 0.862. The flexion obtained by differencing the projection three times at a spacing of six kilometres is 0.871. The two routes share no code — one draws a hundred and sixty-one points and measures a distance, the other evaluates three points and takes a ratio — and they agree to one per cent, which is the third-order term the closed form does not carry.

The consequence for a reader is the same shape as the one the indicatrix is a limit reached one derivative earlier, and it is worth stating in the same words: halving the length of the line halves the error of ruling it, rather than quartering it. The absolute error is quadratic; the error relative to what is being drawn is not.

Mercator’s second-order structure is one elementary function #

On a conformal projection the local scale $\sigma$ is a function of position with no direction in it, which is what conformal means. The only second-order object available is therefore the gradient of $\ln\sigma$, and the two quantities are its two components: skewness along the direction of travel, flexion across it.

Mercator’s scale is $\sec\varphi$, so $\ln\sigma = -\ln\cos\varphi$, and the gradient has length $\tan\varphi$ pointing north. Both extremes are that number:

Measured and predicted agree to five figures at every latitude tested. That is a two-route check on the whole apparatus — the differencing, the geodesic construction and the normalisation — settled by one line of calculus on the one projection whose algebra is short enough to do by hand.

It also settles what the number means. A flexion of one at 45° north is a great circle whose image turns through one radian for each radian of arc, which is a route bending through 57° over a leg of 3,900 kilometres. That is not a subtlety; it is why an aircraft’s track on a Mercator chart looks nothing like the line an atlas reader would draw between the same two cities.

One projection has no flexion, and it is not a coincidence #

If bending is a property of the second derivative, a natural question is whether it can be made to vanish. It can, exactly once.

The measurement is unambiguous. The gnomonic’s flexion at the test point is 4.6 × 10⁻¹⁰, which is differencing noise; the smallest flexion any other projection in the library manages at the same point is 0.168, for the conformal conic. The ratio is 3.6 × 10⁸, so the assertion that the gnomonic is straight and nothing else is has a margin of eight orders of magnitude rather than a threshold somebody chose.

Beltrami’s theorem says this is the whole list: a map taking every geodesic of a surface of constant curvature to a straight line is the gnomonic one, up to a projective transformation. So zero flexion is available, it is available once, and the gnomonic companion is the essay about what it is used for.

What it costs is the subject of the next rung, and the short version is that the gnomonic’s skewness along those same straight lines is $2\tan\rho$, which is unbounded as the mapped region approaches a hemisphere. Straightness is purchased, not found.

The field over a whole map #

The two numbers are defined at a point and in a direction, so a map’s second-order behaviour is a field rather than a number. Reducing it to something that can be drawn means choosing: the largest over directions, or the root mean square over them.

An equal-area projection has an areal factor of exactly one at every point, which is a first-order statement and is true. The same projection bends geodesics by more than a radian per radian in the outer thirds of the map. Neither quantity is wrong; they are measuring different derivatives, and a projection is entitled to be perfect in one and poor in the other.

What this does to the site’s own habit #

The rule this collection is built on is that a projection’s property is computed from its derivatives before the word for it is printed. That rule has been kept, and it has been kept with one derivative.

Three of the site’s standing claims now carry a qualification:

1. Conformal means angles at a point are preserved. It does not mean that a shape drawn on the map has the right form, and it does not mean that the straight line between two places on the map is anywhere near the route between them.
2. An equal-area map’s areal factor of one is a first-order property of infinitesimal patches. The bending of the boundary of a finite region is a separate matter, and computing an area needs a surface is where the finite version of that argument is made.
3. A distortion ranking is a ranking against a stated criterion, and every criterion this site has used — Airy’s, Kavrayskiy’s, the scale spread, the worst ω — is built from first-order quantities alone. Which projection is best is the essay that refuses the question without a stated objective, and it can now be refused one derivative more sharply.

The third of those is the one with consequences, because a ranking is what a cartographer actually uses, and it turns out that ranking the same eight world maps by their second derivative moves Mercator up three places. That is the third rung of this ladder.

Where the model stops #

Two limits are worth naming.

The first is that flexion and skewness are defined along geodesics, and a reader ruling a line on a map is not usually thinking about geodesics. What that reader is doing to the ground is measured at working scale in a straight segment is a claim about a plane, and in the air in flying a curve in straight legs. The bow measured above is the bow of the shortest route away from the chord; a rhumb line, which is the other curve a navigator cares about, has its own bending and it is not this one. Why Mercator exists is about the projection on which one of those two is straight, and the answer is that it is the other one.

The second is that this is the second term and not the last one. A third derivative exists, is not zero, and describes how the bending itself changes along the route. Nothing here measures it. The reason to stop at two is not that the series terminates but that the second term is the first one that contains the failure a reader can see with a ruler, and the site had been printing confident first-order verdicts without it.

Who found it, and when #

Tissot published the indicatrix in 1881, and it has been the standard apparatus for describing map distortion ever since — deservedly, because it is exactly right about what it describes.

The second-order pair is recent. David Goldberg and J. Richard Gott proposed flexion and skewness in 2007 as part of an argument that world maps should be scored on how much they bend and lopside continents rather than only on how much they stretch them, and used the pair to rank the standard collection. Their scoring is a choice among many, and this ladder’s third rung takes it apart on exactly that ground.

What is not in dispute is the arithmetic. The gradient of the log of the scale is available on any projection anybody can write down, the differencing needs three evaluations, and the measurement was there to be made for a hundred and twenty-six years.

One thing about that hundred and twenty-six years is worth stating plainly, because it is a pattern rather than an accident. The arithmetic was elementary throughout: three evaluations of a function anybody could write down, differenced. What was missing was not a technique but a reason to look — the first-order picture answered the questions the subject was asking, so the second derivative was not an unsolved problem but an unasked one. Gaps of that kind do not close when somebody becomes able to do the computation; they close when somebody asks a question the existing apparatus cannot answer, which here was a question about lines rather than about points.

The same reading applies to several of the gaps this collection has found, which is why it is worth naming here.

It also fixes what the second derivative is not. It is not a refinement of the first, in the sense of a correction term that improves an estimate: the flexion of a map is zero on some projections whose angular deformation is large, and large on some whose angular deformation is small, so no amount of first-order information predicts it. The two quantities are independent measurements of independent failures, which is why a ranking on one is silent about the other and why quoting a single distortion number for a projection was always an editorial act rather than a summary.

Which is why the honest way to report a projection is a vector of measurements rather than a score, and why this collection reports several numbers per projection and refuses to combine them. Combining is a decision about what a reader is for, and it belongs to the reader.

None of which makes the first derivative less useful. It makes it one instrument of two, with a stated range.

Where the ladder goes next #

The two quantities have so far been treated as a pair of numbers at a point. They are not independent, and the way they fail to be independent is a theorem with a two-line proof that the machinery found before the essay was written: on a conformal map the two are components of one vector, so their extremes are the same size and sit exactly ninety degrees apart, and conformality removes the second-order freedom as well as the first-order one. On every other projection they are two separate failures with two separate directions. That is the next rung.