An area on the grid is not an area on the ground

A grid’s scale factor is quoted as one number and applied to distances. The scale factor of a line shows that even for a distance it is a subtler object than it looks — a property of a point, applied to something that is not at a point, with three rules in use for averaging it.

What a surveyor is usually asked for is not a distance. It is an area: a parcel, a lease, a planning application, a subsidy claim, a conveyance. And the areal scale factor is not the scale factor.

The identity, and it is checked rather than assumed #

For a conformal projection the areal scale factor is exactly the square of the linear one, because conformality means the scale is the same in every direction and an area is a product of two lengths.

That is an argument, and this site does not print arguments as measurements. Both quantities are computed independently — $k$ from the grid’s own point-scale formula, and the areal factor from the determinant of the Jacobian of the projected map against the ellipsoid’s metric — and required to agree. They agree to $1.2\times10^{-11}$.

The check is worth having because it is a check on the grid rather than on the algebra. A grid that was not conformal would break it, and the transverse Mercator series a national grid actually computes is a truncation whose conformality is a numerical claim rather than an exact one.

Squaring a small departure doubles it #

A hectare at the British grid’s central meridian is short by 797.3 parts per million in area, against 398.7 in length. Doubling, to the accuracy the second-order term allows: $(1+x)^2 - 1 = 2x + x^2$, and $x^2$ here is $1.6\times10^{-7}$, so the doubling is exact to 3.5 parts in ten thousand of itself.

That factor of two is the entire practical content of the essay, and it is the thing most likely to be got wrong in a spreadsheet:

Eight square metres on a hectare is not a rounding. It is a strip a metre wide down the long side of a football pitch, and on a hundred-hectare holding it is 798 square metres — a tenth of a hectare, which is a plot.

Using the line factor on an area leaves half of it #

The mistake this essay exists to name is dividing an area by $k$ instead of by $k^2$. It is exactly the right operation performed once instead of twice, and it leaves half the error behind: on a one-hectare parcel at 55° north, correcting with the line factor leaves 3.49 square metres where correcting properly leaves nothing.

Half is a bad amount to leave. A correction applied and left half-done is worse than one not applied at all in one specific way — the practitioner now believes the number is corrected, and the residual has the same sign and half the magnitude, which is exactly the shape that survives every plausibility check.

The elevation squares too #

The ground is not the grid establishes the other half of a combined factor: a tape on a hillside is reduced to the ellipsoid, and the reduction is a factor $R/(R+h)$ that gets smaller as the ground rises.

For an area that factor squares as well, and it enters with the opposite sign to the grid’s. Ground higher up is further from the centre and there is therefore more of it under a parcel of given angular size:

The extra 63 parts per million is $2h/R$, and it is negative — which is a thing worth stating carefully, because the sign is the one part of this that is easy to get backwards. The site’s own machinery had it the wrong way round until the two expressions in the same function were compared: one made the elevation term more negative and the other less, and only one of them can be right.

Where the zone crosses over #

A grid with a scale factor below one is too small near its central meridian and too large near its edges, so somewhere in between it is exact. For lengths that happens where $k = 1$; for areas it happens at the same place, because $k^2 = 1$ exactly when $k = 1$.

That coincidence is not trivial and is worth noticing. It means a survey that has located the line on which its plan’s distances are true has also located the line on which its areas are true, and there is no second calculation to do. Everywhere else the areal departure is twice the linear one, so the band within which areas are good enough is narrower than the band within which lengths are — by a factor of $\sqrt{2}$ in distance from the crossover, since the departure grows quadratically with easting.

Everything in this essay is the grid’s own scale curve squared — the same shape across the zone, the same zeros, and twice the departure at every easting. A tolerance on area therefore fixes a narrower usable strip than the same tolerance on length, and no grid on record has been designed against the area one.

The shape of the parcel does not matter, and that is not obvious #

Everything above is computed for a square parcel, and a reader is entitled to ask whether a long thin one behaves differently — the scale factor varies across a zone, so a parcel stretched east–west spans more of that variation than a compact one.

It does not matter to first order, and the reason is that the areal factor is a pointwise quantity being integrated. A parcel’s true area is $\iint k^{-2}\,dA$ over the parcel, so what enters is the mean of $k^{-2}$ over the parcel rather than its value at the centre — and the difference between those two is second order in the parcel’s size, because $k^2$ is smooth.

For a hectare that difference is negligible by any measure: the parcel is 100 metres across and $k$ changes by 0.03 parts per million over that distance at the zone’s steepest. For a parcel a hundred kilometres across it is not, and the correct operation there is an integral rather than a factor — which is exactly the situation the scale factor of a line describes for distances, with Simpson’s rule as the answer and its fourth-order convergence asserted.

So the rule for areas is the rule for lines with the exponent changed: a factor at the centroid for a small parcel, an integral for a large one, and the crossover set by a stated tolerance rather than by the parcel’s name.

Which of the two a specification names #

Almost always the length. National grid specifications, survey standards and setting-out tolerances are written in parts per million of distance, because distance is what an instrument reads.

The consequence is a silent factor of two whenever the deliverable is an area. A job specified to 200 parts per million on distance is a job at 400 on area, and the specification does not say so. That is the same structure as the tolerance decides the model — a stated tolerance implies which corrections may be dropped — with the twist that the implication depends on what is being sold, and the specification does not know.

The honest form of a specification for a parcel is a tolerance in square metres or in parts per million of area, and converting a distance tolerance to it is a halving rather than a copy.

What the legal quantity actually is #

Nothing here says which area is the right one to sell, and that is deliberate.

A conveyance may describe a parcel by its grid coordinates, in which case the grid’s area is the definition and the ground’s is irrelevant. It may describe it by monuments on the ground, in which case the ground’s area is the definition and the grid’s is an artefact of how it was recorded. It may state an area, in which case the two differ by the numbers above and something has to give.

Jurisdictions answer this differently and this site does not take a view — the cadastral half of the subject is reserved, as the practice field’s own claim says. What the geometry contributes is the size of the difference and the fact that it is not zero, which is the part a legal argument cannot supply for itself.

The site grid makes it worse in one direction #

A construction site works in coordinates scaled so that a tape reading equals a computed distance, which a grid scaled to the ground is not a map measures: the result is exact at one point and degrades east–west forty times faster than north–south.

For areas that anisotropy doubles as well, and it doubles in a way the linear case hides. A site grid’s departure is no longer isotropic — one direction is much worse than the other — so its areal factor is no longer the square of a single number but the product of two different ones. The parcel’s area error then depends on its orientation as well as on its position, which is a genuinely new failure mode: two identical rectangles at the same place, one aligned north and one aligned east, have different area errors.

That does not happen on a proper grid, because a proper grid is conformal and a conformal map has one scale factor per point. It happens on a site grid precisely because a site grid is not a projection.

Two grids over the same parcel #

A parcel recorded on two different grids has two different plan areas, and the difference is not the difference between the grids’ scale factors — it is twice it.

Two grids over the same ground separates the five parameters of a grid definition and finds that only one of them changes the map. For a length that one parameter’s effect is the difference in $k$; for an area it is doubled, so a parcel transcribed from the British grid to UTM changes its recorded area by twice whatever the scale factors differ by at that place, plus twice whatever the two ellipsoids differ by.

That is a real transcription hazard rather than a theoretical one. A parcel is recorded once and cited for decades, and a jurisdiction that changes grid inherits a set of areas that are all wrong by a systematic factor nobody notices because every one of them is wrong by the same amount.

The other places the square appears #

Three, and they are worth listing because the doubling is the same doubling each time.

A density. Anything per unit area inherits the areal factor, so a population density computed from grid areas is out by the same 800 parts per million, in the opposite direction.

A volume. An earthworks quantity is an area times a depth, and the depth is a height rather than a grid distance, so a volume carries the areal factor once and not the linear factor three times. That is the mistake available in the other direction.

A rate of change. A gradient is a height over a grid distance, so it carries the reciprocal of the linear factor — one power, not two — and a slope computed from grid coordinates is out by 399 parts per million rather than 797.

Each of those is one line of dimensional analysis, and each of them is wrong in published spreadsheets, because the scale factor is remembered as a number rather than as a quantity with an exponent attached to it.

A worked chain, end to end #

Putting the pieces in order for one parcel says what a practitioner actually has to do, and how many steps there are.

A hectare is set out on the ground at 52° north, 300 metres up, half a degree east of the British grid’s central meridian. The tape reads 100 metres on each side, so the ground area is exactly 10,000 square metres by construction.

1. Reduce to the ellipsoid. Each side shrinks by $h/R = 47$ parts per million, so the area shrinks by 94: 9,999.06 square metres.
2. Project onto the grid. Each side is multiplied by $k$, which at half a degree from the central meridian is 0.99961; the area is multiplied by $k^2 = 0.9992316$: 9,991.38 square metres.
3. Read it off the plan. The plan says 9,991.4 square metres and the ground holds 10,000.

Eight and a half square metres, on a parcel whose sides were measured to the millimetre. The chain has two steps, each of which squares, and both are in the same direction here because the grid is below unity and the ground is above the ellipsoid.

That is the areal version of the four-step chain what a tape measures sets out, and the ordering matters for the same reason: the two factors do not commute with the projection, and applying them in the wrong order costs the cross term.

The exponent for a quantity that has no integer dimension #

The ladder’s closing note defers the case where the exponent is not a whole number, and it is worth supplying, because the arithmetic is short and the answer is not the one the deferral implies.

A grid is locally a similarity of the ground with ratio k, so a curve on the grid is the ground curve scaled by k. Measuring the grid curve with a ruler of r grid units counts the same steps as measuring the ground curve with a ruler of r/k ground units, and for a curve of fractal dimension D whose measured length goes as s^(1−D),

$$L_{\text{grid}}(r) = k\,L_{\text{ground}}(r/k) = k^{D}\,L_{\text{ground}}(r).$$

The exponent is the fractal dimension. Not one, and not two: D, which for a coastline is somewhere between 1.1 and 1.3.

So a coastline whose length was measured on the British grid at a fixed number of grid units per step, at the central meridian, is short by 399 × D parts per million rather than by 399 — 503 parts per million at D = 1.26, which is a quarter more correction than the linear rule gives. A thousand kilometres of coast recorded that way is out by 503 metres where a naive grid correction removes only 399.

Three things are worth taking from that.

The rule generalises the essay’s own exponent table rather than contradicting it. A smooth curve has D = 1 and the correction is k, which is the row already there; an area-filling curve would have D = 2 and the correction would be k², which is the area row. Every entry in the table is this expression evaluated at that quantity’s own dimension, and the integer cases are the ones where the dimension happens to be an integer.

And the correction needs a number the data does not carry. D is a property of the coast, measurable only over a stated range of scales, and it is exactly the quantity a line has a length only at a scale shows cannot be read off a single measurement. So the correction for a coastline’s grid-to-ground length is knowable in form and not in value, which is a worse position than any other row of the table.

The practical escape is to fix the ruler on the ground rather than on the grid. A length measured with a step of a stated number of metres of ground has no D in its correction at all — it is a sum of ground distances and needs none. The exponent appears only because the ruler was in the plane, and moving the ruler off the plane removes it, which is the same repair every other rung of this ladder recommends for a different quantity.

Where this ladder goes #

The grid ladder has established what a grid declares, where its origin is, what its scale factor is and why it was chosen, what happens to a line, what happens at a seam, and what a site grid scaled to the ground is.

This adds the exponent: every quantity computed on a grid carries the scale factor raised to the power of its own dimension in length, and the specification names the power one case.

What is left on this ladder is the case where the exponent is not an integer — a quantity like a perimeter-to-area ratio or a fractal dimension of a coastline, where the scaling is genuinely mixed — which belongs to whoever owns shape rather than to a grid.