Total curvature and the scale rule

No map is faithful is a statement with no number in it. This essay puts one there.

The distortion a map must impose is bounded below by the curvature it has to absorb, and for one important case the bound can be written down exactly rather than estimated — which turns “projection choice matters more for larger areas” from a rule of thumb into arithmetic.

The quantity that is fixed #

Gaussian curvature is a local quantity, and its integral over a region is not affected by anything a projection can do.

For a region $\Omega$ on a sphere of radius $R$, the total curvature is

$$\int_\Omega K\,\mathrm{d}A = \frac{\text{area}(\Omega)}{R^2}$$

because $K = 1/R^2$ everywhere. A region covering a fraction $f$ of the sphere therefore carries total curvature $4\pi f$.

That number is not a property of any map. It is a property of the region, and it is what any flat picture of the region has to absorb. A projection can decide where the distortion falls; it cannot reduce the total it is working against.

The scale rule #

The consequence is a rule with orders of magnitude in it:

Six orders of magnitude separate a city from a continent, and the distortion any map must impose scales with them. That is why a surveyor treats a site as flat without apology and a world map cannot be treated as anything.

The transition is not gradual on a human scale. Below about a kilometre the forced distortion is below any instrument; above about a thousand kilometres it dominates every design decision; in between there is a band where it matters and is manageable, which is where national grids live.

The bound, exactly #

The curvature integral says how much there is. It does not say what the best possible map does with it, and for one shape of region that second question also has a closed answer.

Take a spherical cap of angular radius $\rho$ — the region within $\rho$ of a chosen point — and ask for the conformal map of it whose scale varies least. Chebyshev’s criterion names the winner: the projection whose scale is constant on the boundary. By symmetry that is the stereographic projection centred on the cap, and its scale factor at angular distance $c$ from the centre is $\sec^2(c/2)$. So the ratio between the largest and smallest scale over the cap is

$$\frac{k_{\max}}{k_{\min}} = \sec^2\frac{\rho}{2}$$

That is a lower bound, not the behaviour of one projection. No conformal map of that cap does better, and every other one does worse.

A hemisphere’s scale must double. A continent-sized region must vary by seven per cent. A survey site a kilometre across must vary by 7.6 parts per billion, which no instrument has ever resolved.

The curvature integral is directly measurable, which is what makes the bound usable rather than merely true. The angles of a spherical triangle overshoot 180° by exactly the area it encloses, so a region’s total curvature is available from measurements taken on its boundary alone.

The two statements are the same statement #

Worth checking, because a bound and an integral are different kinds of object and their agreement is not obvious.

For a small cap, $\sec^2(\rho/2) \approx 1 + \rho^2/4$, so the forced excess scale ratio is $\rho^2/4$. The total curvature of the same cap is $2\pi(1-\cos\rho) \approx \pi\rho^2$.

Their ratio is $4\pi$, independently of $\rho$. The forced distortion is proportional to the enclosed curvature, with a constant that is the same for every small region — and the site asserts exactly that, requiring the ratio to stay within two per cent of $4\pi$ for caps from half a degree to five degrees.

So the theorem and the engineering rule are not two facts that happen to point the same way. They are one fact in two units.

What the bound does not cover #

Three limitations, and they are the reason this is one exact case rather than a general theory.

It is for conformal projections only. Give up conformality and the scale ratio can be reduced further — an equal-area projection of a cap has a smaller scale ratio and a large angular deformation instead. The bound constrains one family.

It is for a cap. A region that is long and thin has a much smaller total curvature than a cap of the same diameter, and its optimal projection is not stereographic. Chebyshev’s criterion still applies and still names a unique answer; that answer just has no closed form for a general boundary, and finding it is a genuine boundary-value problem.

It is a bound on scale variation, not on distortion in general. Two projections with the same scale ratio can distribute it very differently, and a map that is 7% out uniformly and one that is exact over 90% of its area and 7% out in a corner have the same number attached.

The third is the sharpest limitation and it is why the regional criteria exist: a single ratio is a worst case, and a worst case is not a description.

Why a long thin region is cheap #

The cap is the worst case for a region of a given diameter, and understanding why explains most of what good projection choice consists of.

Total curvature is proportional to area, not to extent. A region five degrees across in both directions is a cap of radius 2.5° and carries curvature $\pi\rho^2 = 5.9\times10^{-3}$. A region fifty degrees long and one degree wide has roughly the same area — and therefore roughly the same total curvature — while spanning twenty times the distance.

So a country that is long and thin can be mapped with far less distortion than its extent suggests, provided the projection’s zero-distortion line runs along its length. Chile spans 39° of latitude and 5° of longitude; a transverse cylindrical projection through it absorbs the curvature of a strip rather than of a cap fifteen degrees across, and the difference is more than an order of magnitude.

That is the quantitative form of the advice that the aspect should be chosen to fit the region. It is not merely tidier. A projection whose zero line runs the wrong way across a long thin country is being asked to absorb the curvature of the region’s bounding cap rather than of the region, and the bounding cap of a long thin shape is enormous compared with it.

The site’s own regional ranking bears this out sharply. Over Europe — wide, shallow, mid-latitude — a conformal conic scores 0.042 on the Kavrayskiy criterion and Mercator 0.229. Over Chile the order reverses completely: Mercator 0.147, the conic 0.307. The same six projections, ranked in different orders over the two regions, and the reason is entirely which way the zero line runs relative to the shape.

A national grid works comfortably inside the bound for the same reason. Across one zone at 50° north the measured scale varies by well under a part in a thousand — far less than the 1,900 parts per million a cap of Britain’s diameter would force, because Britain is not a cap.

What the bound says about a national grid #

A worked case, because the numbers land somewhere useful.

Britain fits inside a cap of about 5° radius. The bound says any conformal map of it must vary in scale by at least $\sec^2(2.5°) = 1.0019$ — 1,900 parts per million between the best and worst point.

The British national grid, a transverse Mercator with a scale factor of 0.9996 on a central meridian at 2° west, achieves a scale range of roughly 400 parts per million too small on the axis to about 500 too large at the extremities: a ratio of about 1.0009.

That is better than the bound, and it is not a contradiction. The bound is for a cap of 5° radius, and Britain is not a cap — it is long and thin, running north–south, with the grid’s zero lines running the same way. The region’s actual total curvature is a fraction of its bounding cap’s, and the grid is exploiting exactly that.

Which is the practical lesson the bound teaches by being beaten: the forced distortion is set by the region’s area and shape together, and a projection fitted to the shape does better than a bound computed from the diameter. A grid that scored worse than the cap bound would be a grid that had thrown away its aspect.

The rule can also be exploited deliberately. Cutting a world map into lobes gives each piece a fraction of the sphere’s curvature to absorb, and the mean shape distortion falls accordingly — at the cost of the edges, which is the quantity the cutting adds.

That figure is the curvature integral used as a design tool rather than as a constraint. Interrupting a map does not reduce the total curvature of the world; it hands each lobe a smaller share, and the distortion each lobe has to absorb falls with the share rather than with the lobe count.

Where the rule earns its keep #

The practical residue is a way of answering “does the projection matter here” before choosing one.

Compute the region’s angular radius. Below a degree, the forced distortion is under a hundred parts per million and the projection choice is administrative — use whatever the neighbouring data uses. Between one and ten degrees it is between a hundred and eight thousand parts per million, which is where a well-fitted projection is worth having and a badly-fitted one costs real accuracy. Above thirty degrees nothing is good and the question becomes which failure to accept.

That threefold split is what the standard advice about scale amounts to, with the boundaries computed rather than felt.

Three surfaces carry the whole argument. The plane has zero curvature and needs no map; the cylinder has zero curvature and unrolls exactly; the sphere has $1/R^2$ everywhere, and the integral of that over a region is the quantity no projection can touch.

The three regimes, in numbers #

The scale rule is most useful as a threshold test, and the thresholds can be stated rather than felt.

Below about 1° of angular radius — a region a hundred kilometres across — the forced scale variation is under 76 parts per million. That is smaller than the discrepancy between two ellipsoids, smaller than a datum’s internal network distortion, and comparable to what a survey network accumulates over the same distance. At this scale the projection is not the limiting factor and choosing one is an administrative decision: match whatever the neighbouring data uses.

Between about 1° and 10° — a country — the forced variation runs from 76 to 7,700 parts per million. This is the band where a fitted projection is worth having and a badly fitted one costs real accuracy, and it is exactly the range national grids operate in. A well-chosen grid gets close to the bound; a normal-aspect projection applied to a north–south country can be an order of magnitude worse.

Above about 30° — a continent or the world — the forced variation is 7% and upward, reaching 100% for a hemisphere. Nothing is good here, the choice is between failures rather than toward an optimum, and the question stops having a technical answer.

The boundaries are soft and the orders of magnitude are not. Six decades of area separate the first regime from the third, and the same six decades separate a projection choice that cannot matter from one that decides everything about the map.

The other direction #

One thing the bound does not say, and it is a common misreading.

A region with total curvature $C$ must absorb $C$; that does not mean a map of it is guaranteed to be as good as the bound. The bound is what a perfectly chosen projection achieves. A badly chosen one is arbitrarily worse: Mercator over the same 30° cap has a scale ratio of 3.73 against the bound’s 1.07, which is a factor of fifty in the excess.

So the curvature integral gives a floor and says nothing about the ceiling. Both facts are worth having and they answer different questions: the floor says whether a good map is possible, and the gap between the floor and a candidate says whether a particular map is good.

Measuring that gap is the whole of what a regional criterion does, and the bound is what makes the measurement meaningful — a number is only informative against something it could have been.

Total curvature is a topological quantity and does not care how the curvature is distributed. On a body whose curvature varies, the distribution is a second thing worth knowing.

What was computed here #

The curvature integral and the cap bound are both closed forms and both are evaluated directly. The proportionality between them is asserted: the ratio of total curvature to excess scale ratio must stay within two per cent of $4\pi$ across caps from 0.5° to 5°, and it does to 0.2%.

The bound is checked against the projection that is supposed to achieve it. The stereographic projection centred on a cap must have a scale ratio equal to $\sec^2(\rho/2)$ — measured from its own derivatives over a sampled region including the boundary and the centre — and it does, to 1.4×10⁻⁸ at 30°.

That measurement was wrong the first time in an instructive way. The region sampler used equal-area rings whose innermost and outermost sat half a step inside the centre and the rim, so it reported a scale ratio below the proved bound by three parts in a thousand. A quantity that comes out better than a theorem is a sampling artefact every time, and the fix was to sample the extremes where they actually occur rather than near them.

Every rival is required to do worse. If some conformal projection ever matched the bound over a cap without being the stereographic one centred on it, either Chebyshev’s theorem or the measurement would be wrong, and the build stops rather than reporting a tie.

What the pictures cannot show #

The integral. Total curvature is a number obtained by adding up a quantity that has no visual appearance, over a region, and every figure here shows either a surface or a scale profile. The hero figure comes closest — the horizontal line in it is the bound, and it is a consequence of the integral — but the line is a result rather than a picture of the reasoning.

The figures also cannot show a region that is not a cap, which is most regions. The exact bound exists for the symmetric case and the essay says so; for a country the criterion still names a unique optimum and computing it is a different kind of problem.

Who found it, and when #

Gauss’s Theorema Egregium of 1827 makes curvature intrinsic. Bonnet’s 1848 result, building on Gauss, makes its integral over a region equal to a boundary quantity — which is the version this site computes.

Chebyshev stated the optimality criterion for conformal projections in 1856 and did not prove it; Grave supplied the proof in 1896. The closed form for the cap follows from symmetry and was known well before either.

The engineering rule — that projection choice matters in proportion to the extent — has been folklore in surveying for as long as there have been surveys, and it is the same statement. What is unusual is how rarely the two are put side by side, given that one is the other with the units changed.

Two statements, two audiences, one arithmetic #

The remark that the theorem and the folklore rule are the same statement with the units changed is the essay’s real content, and it is worth drawing out why they stayed apart for so long.

They are addressed to different questions. The bound answers what is the least distortion any conformal map of this region can have, which is a question about all possible maps and is the kind of thing a theorem is for. The engineering rule answers does the projection matter on this job, which is a question about one job and is the kind of thing a specification is for.

And they are stated in different currencies. One is an integral of curvature over a region, in dimensionless units, proved once. The other is a threshold in kilometres, calibrated to a tolerance in millimetres, quoted in a manual. Nothing in the appearance of either suggests it is the other.

The people who hold them do not overlap. A surveyor deciding whether to apply a scale factor is not reading Grave’s proof, and a differential geometer computing a total curvature has no occasion to convert it into a working radius. Both are behaving reasonably; the translation between them is work neither needs.

Which is what makes the join worth publishing. The bound tells the surveyor that their rule is not a rule of thumb at all but a theorem with a constant in it, so it can be recomputed for a different tolerance, a different region shape, or a different body rather than inherited. And the rule tells the geometer what the bound is for.

The general shape is one this collection meets repeatedly: a result and a practice that are the same arithmetic, kept apart by their units and their audiences, with the translation available and unwritten because nobody’s job required both halves.

Where this goes next #

The theorem the bound rests on is no map is faithful. The criterion that names the optimal projection is Chebyshev’s criterion. And the way the curvature integral is actually measured is measuring curvature from inside.