The contour is right and the reading is wrong

A projection moves points. A field attaches a number to each point. So a projection cannot change which number is attached to which point, and the set of points carrying a stated value is the same set on every map ever drawn of that field.

That is the one thing about a field a map cannot get wrong, and it is worth saying plainly because it is the reason contour maps work at all. A contour line is an exact object. It is transported by the projection like any other curve, and the value it carries travels with it.

What survives, stated carefully #

The invariants of a projection are usually discussed as quantities that do not depend on the parameterisation — the principal scale factors, the areal factor, the maximum angular deformation. This is a different kind of survival and a stronger one: the value of the field is not an invariant of the projection, it is untouched by it, because it was never a geometric quantity in the first place.

A tempting way to put it is that a projection is a bijection and a field is a function of position, so composing them changes neither. That is right and it is worth being suspicious of, because exactly the same sentence would be true of an area — an area is a function of a region, and a region maps to a region — and areas plainly do not survive.

The difference is where the arithmetic happens. A value is attached to a point and read off at a point, so the projection has no opportunity to act on it. An area is computed from the coordinates, and the coordinates are what the projection changed. Everything in this essay is a quantity of the second kind wearing the clothes of the first: it looks like something read off the contours, and it is something computed from where they landed.

The hypsometric curve, exactly right on one map #

The hypsometric curve is the oldest thematic summary in physical geography: the fraction of a surface lying above each elevation, and the shape that lets somebody say a continent is mostly lowland or mostly plateau. It is an area statistic of a field, so it is exactly the case an equal-area projection was built for.

Measured over the same field on the same grid, the equal-area map’s curve and the sphere’s own agree to 5.0 × 10⁻¹³ of the total area — which is what “equal-area” means, exercised on something other than a cell. The conformal map’s curve departs by up to 10.8 per cent of the total.

The departure is not random and it is not small in the places anybody cares about. Mercator inflates high latitudes, so any value that occurs disproportionately near the poles is over-represented in the curve and every value that does not is under-represented. A reader taking the hypsometry off a Mercator sheet gets a systematically wrong distribution and has no way to see it, because the contours themselves are perfect.

That is the reverse of the slope reading, where the conformal map is the one that is right about direction and the equal-area map is the one that is wrong. The same pair of maps, the same field, two readings, and the correct choice is opposite in the two cases.

What the spacing says, and what it means #

The most common reading of all is the spacing: contours close together mean steep ground, far apart mean gentle. That reading is a gradient measurement made with a ruler, and it inherits everything the previous rung established.

The important word is along one contour. A reader who knew the map’s scale factor varied could in principle correct for it — but the correction would have to be applied point by point along a single line, and the whole convenience of reading spacing is that it is done by eye across a whole sheet.

On the conformal map the ratio is exactly 1/k at every point, so the spread is a spread of the scale factor and nothing else: 0.4899 to 1.0000 over the sampled points. On the equal-area map the ratio is not a function of position alone — it depends on which way the contour happens to run — and it spans 0.8106 to 2.0409.

The length of a contour, which is neither #

A contour’s length on the page divided by its length on the ground is a third quantity, and it is not the same as either of the first two.

Over four levels of the same field, the ratio of page length to ground length runs 1.11, 1.17, 1.25 and 1.67 on a conformal map and 1.05, 1.08, 1.12 and 1.40 on an equal-area one. Neither is constant, so a reader who measured a contour with a piece of string and converted with the map’s stated scale would be out by a factor that depends on which contour it was.

The pattern is worth reading. The highest level in the set — the one that closes around the summit at the top of the field — has the largest ratio on both maps, because it is the shortest and most northerly of them and lies where both projections stretch the most. That is what makes it a measurement of the map rather than of the field: the same physical curve gets a different answer depending on where it sits.

Two contours and the ground between them #

A quantity that sits between the two extremes is worth its own paragraph, because it is the one most thematic maps actually print: the area lying between two neighbouring contours, which is the band a hypsometric tint fills.

It is an area, so an equal-area projection returns it exactly and a conformal one does not. That much follows from the curve above. What is less obvious is that the error is not uniform across the bands. On a conformal map the areal factor is k², so a band lying at high latitude is inflated by the square of a number that is already large, while a band lying near the equator is barely touched. A reader comparing two tinted bands of a Mercator sheet is comparing one that has been multiplied by four with one that has been multiplied by one — and both bands are bounded by contours that are exactly correct.

That is the mechanism behind the oldest complaint about the Mercator world map, which is usually stated about countries. Stated about a field it is sharper, because the bands are not arbitrary regions chosen by history: they are level sets of the thing being mapped, so the comparison a reader is invited to make is exactly the comparison the projection has corrupted.

The same argument run the other way explains why an equal-area map is the wrong sheet to read a slope from even though it is the right one to read an area from. The band’s width is what the eye converts to steepness, and a width is a length, and the length of the perpendicular between two contours is stretched by whichever principal scale factor happens to lie across the band. On an equal-area map that factor swings above and below one across the sheet, and a band of constant true width is drawn wide in one place and narrow in another.

What was computed, and how #

The contours are traced by marching a longitude–latitude grid and interpolating linearly across every cell edge where the field crosses the level. That gives an unordered set of segments, which is all that any measurement here needs: a length is a sum over segments and an area is an integral over a region, and neither needs the contour walked in order.

Two checks make the tracing trustworthy rather than plausible.

The first is that every traced point carries the value it was traced for. Evaluating the field at the traced points and comparing with the level gives a worst departure of under 5 × 10⁻⁴, which is the linear interpolation’s own error across one grid cell and falls as the grid refines.

The second is the assertion this essay’s claim rests on, and it is deliberately trivial to state: after projection, the value at each traced point is identical to the last bit, because projection does not touch the value. An assertion that can only pass is normally worthless. This one is here because the reader’s intuition says the opposite — that a projection must do something to a contour map — and the point is that the thing it does is entirely to the measurements.

The hypsometric curve is computed twice over the same sample points, once weighting each point by the sphere’s area element cos φ and once by the page’s, which is the Jacobian determinant of the forward map. Both use the same field values and the same points, so the difference between the two curves is what the projection did to area and nothing else.

The ground lengths are computed on the sphere, segment by segment, with the great-circle distance. The page lengths are Euclidean in the projected plane. Comparing them needs no scale convention because the projections here work in units of the sphere’s radius, which is the same convention the areal factor is computed in.

The measurement that would settle it, and what it costs #

A reader who wanted both readings from one sheet has a real option, and it is worth pricing rather than dismissing: print the correction. A conformal map’s scale factor is a known function of position, so a graticule annotated with k at every intersection lets a slope reading be corrected exactly, and an equal-area map’s areal factor is one everywhere so no correction is needed for the hypsometry.

That is what a large-scale national sheet does for distances, in the form of a stated scale factor for the grid, and it works because such a sheet covers a region small enough that the factor varies by parts per million. On a world map the factor varies by a factor of five over the drawn extent, and a correction table for it is not an annotation any more — it is a second map.

The honest position is the one this collection keeps arriving at: the choice of projection is a choice of which measurement will be right, and printing the numbers a reader would need to undo it is only a way of admitting which choice was made.

Where the model stops #

A contour drawn from data is not the level set. Everything above is about the exact level set of an analytic field. A real contour map is interpolated from samples, and its lines are wrong by an amount that depends on the sampling — which is the error an edge in a raster suffers and is not a projection question at all. The claim being made is that projection adds nothing to that error, not that the error is zero.

Contour intervals are chosen, and the choice is not neutral. A map with a 10-metre interval and one with a 50-metre interval of the same ground carry different impressions of steepness, and nothing here bears on that. The measurements above hold the levels fixed and change only the projection.

The hypsometric result is about the map’s own frame. The 10.8 per cent is a fraction of the area the conformal map draws, over a grid reaching 78° of latitude. Extending it further inflates the number without limit, because Mercator’s areal factor has no finite bound — so the honest quotation is the one with the extent stated, which is why it is stated.

The generalisation #

Three readings, three answers, one field:

• the value at a point survives every projection exactly;
• the area between two contours survives an equal-area projection exactly and no other;
• the spacing, and therefore the slope survives a conformal projection up to a single known factor, and no other.

No map does all three, and the reason is the same two lines that forbid an isometry: a map that got both the areas and the gradients right would have ab = 1 and a = b, hence a = b = 1, which is the isometry the Theorema Egregium refuses.

So a thematic map of a field is not one map. An atlas plate that prints a hypsometric table beside a contour drawing on a single sheet is printing two quantities that want two different projections, and at least one of them is being read off the wrong one.

The one reading that is a trap #

There is a fourth reading, and it is the one that catches people who know about the first three: the shape of a contour. A closed contour that is circular on the ground and elliptical on the page has been distorted, and a reader who has learnt to distrust areas and lengths might reasonably think shape is safe on a conformal map.

It is not, and the reason is already on this site. Conformal does not mean the angles are right at any finite size: conformality is a statement about the limit as a figure shrinks, and a contour a thousand kilometres across is not in that limit. The distortion of a finite closed curve on a conformal map is governed by the variation of the scale factor along it, and on a world map that variation is large.

So the ordering is: the value is exact, areas are exact on one kind of map, gradients are exact up to a known factor on the other kind, and shape is exact on neither at any size a contour actually has. Four readings, and the only unconditional one is the one nobody thinks to doubt.

Who found it, and when #

Contour lines are older than the theory of them: Nicholas Cruquius drew depth contours of the Merwede in 1729 and Marsigli had drawn isobaths in the Gulf of Lion in 1725. Contours of terrain arrived with Dupain-Triel in 1791 and became the standard of military survey in the nineteenth century.

The hypsometric curve is Alexander Supan’s, from the 1880s, and it was published as a table of areas — which is to say that the quantity it reports was understood from the start to be an area statistic rather than a picture. The practice of reading it off a projected map came later and quietly.

The observation that contour spacing is a gradient measurement is in every field manual. The observation that this makes the choice of projection for a contour map opposite to the choice for a chloropleth of the same field does not appear to be, which is what happens to a fact that lives between two specialisms.

That opposition has a consequence for anybody producing a single sheet. A physical map showing relief by contours and a thematic map shading a field by area want different projections, and a general-purpose atlas plate that does both has silently chosen which of its two readings is the trustworthy one. Nothing on the sheet says which, and a reader who takes a gradient off one plate and an area off the next is right about one of them.

Where the ladder goes next #

Reading a contour map is sampling the field’s gradient at scattered points. Integrating it is a different operation, and it is what every drainage analysis, flow accumulation and watershed delineation does: follow the direction of steepest descent from a point and see where it ends up.

Because a conformal map preserves that direction exactly, it preserves the whole trajectory — and because an equal-area map does not, it sends the water somewhere else, by a distance that this collection can measure.