Scale factor — where it appears
Named by 59 essays across 8 fields — each of them below, with the objects they name alongside it.
A screen map is a pyramid of tiles
The scheme every slippy map runs on is a coordinate system with three integers and one projection, and almost all of it is forced. A square world is what makes the quadtree work, the levels are exact powers of two, and the published resolution — 156,543 metres per pixel at zoom zero — is a distance on the ground at exactly one latitude.
Tissot's indicatrix
A tiny circle on the sphere becomes an ellipse on the map, and the ellipse's two axes are the whole story. Almost every published indicatrix is drawn without them, which discards the content and keeps the decoration.
A grid has an origin that is not there
Every published national grid measures from a point in the sea. The British one sits 400 kilometres west and 100 kilometres south of where the projection's own zero is, and moving it changes every coordinate in the country and not one distance or bearing.
What a standard parallel buys
A standard parallel is a line where the projection is exact. Choosing one does not reduce the distortion — it decides where the distortion is zero and lets everything grow away from it.
What survives a change of coordinates
The scale along the meridian is a property of the map and the grid together. The principal scale factors are a property of the map alone. Only the second kind describes the projection, and the two are routinely quoted as though they were the same thing.
The seven parameters, and what each one does
A datum transformation is published as three metres, three arcseconds and one part per million, in three different units — which hides the fact that all seven are the same size of thing. Converted to ground displacement, the scale term beats two of the translations.
The scale of a screen map is not one number
A zoom level prints one scale for the whole world, and the map is at that scale along exactly one line. At 60° north the picture labelled 1:136,495 is a 1:68,765 map — and the factor is not quite sec φ either, because the projection puts a geodetic latitude into a spherical formula.
What a grid is made of
A projected coordinate declares five things — a datum, a projection, its parameters, a unit and an axis order — and four of them are invisible in the numbers. Every one has a documented case of being guessed wrong, and the cheapest of those cost a hundred metres.
The scale factor of a line
A scale factor is a property of a point and a distance is not measured at a point. The three rules in use for averaging one along a line differ by a hundred metres on a 362-kilometre baseline, and which of them is adequate is decided by the line's direction rather than its length.
A scale bar is right in one place
The bar in the corner of a world map is a picture of a distance, and it is a true picture along one line. On Mercator it reads 500 kilometres for a thousand at 60° north — and on an equal-area map it reads 500 one way and 2,000 the other, so the projection recommended for measuring is the one on which no single correction exists.
The two ways a map is wrong
Angles and areas fail independently. A projection can be perfect about one and catastrophic about the other, and a single number summarising "how distorted" a map is has already thrown away the distinction that matters.
The scale factor was chosen
A grid's scale factor is not a constant of nature. Britain's 0.9996012717 is the reciprocal square root of the worst distortion a tangent projection would have had over the country, it halves the worst-case error exactly, and the halving is the most that construction can ever buy.
The square costs the poles
The cut at 85.0511287798° is the most-quoted number in web mapping and is almost never derived. It is where Mercator's northing equals half the world's width — the condition for a square — and it drops 1,901,487 square kilometres. A two-tile root would have reached 89.786° and dropped 3,558.
Transverse Mercator and the series that computes it
The projection most of the world's survey data lives in has no closed form. What every national grid actually computes is a truncated power series in the flattening, and how far it can be trusted is an engineering parameter rather than a property of the projection.
Measuring instead of naming
A projection is called conformal because that is its name. Running the definition as a computation over several hundred points takes about twenty lines, catches a projection the whole internet uses, and is almost never done.
Chebyshev's criterion
The only optimality theorem in the subject. Among all conformal projections of a region, the one with least scale variation is the one whose scale factor is constant on the boundary — a criterion with a proof, a unique answer, and a test the site can run.
UTM and the zone system
Sixty separate maps of the world, each six degrees wide, each with a scale factor of 0.9996 chosen so the projection is wrong everywhere and less wrong at the edges. Every constant in the definition is a measured trade rather than a convention.
The pixel is a place with a size
Drawing a coordinate rounds it to a pixel, which moves it by up to half a diagonal — 16.8 metres at zoom 12. Whether two points 43 metres apart appear as two dots is not a property of the data at all: they merge for 12 per cent of the positions the tile grid could take, and the closed form predicts 12.4.
Scale distortion is the third failure
A map's stated scale is its scale at one place. Everywhere else it differs, by a factor that no conformal projection can avoid and that neither of the two usual distortion measures reports. It is the failure everybody uses and nobody counts.
The units are part of the coordinate
Two feet were in use in the United States until 2022, differing by exactly two parts per million. A plan drawn in the wrong one has every dimension right to a fifth of a millimetre and sits three-quarters of a metre from where it belongs, which is why it passes every check anybody runs.
Designing a grid for one region
A country with a grid of its own beats the international zone that would otherwise carry it by a factor of 1.51. That is worth having and it is much less than the argument is usually made to sound, and the difference between the two numbers is the whole case for and against national grids.
How small is flat enough
A builder works in plane coordinates and a national mapping agency does not, and the line between them is not a convention. The unavoidable error of treating a patch of the Earth as flat grows as the square of its size, and the size at any stated tolerance is a number.
The pyramid did not have to be Mercator
The usual defence is that a quadtree needs a square world and Mercator supplies one. So does the cylindrical equal-area with standard parallels at ±55.654° — the solution of π cos²φ₀ = 1 — and it needs no polar cut at all. What Mercator actually buys is conformality, and the price of giving it up is 13.8° of shear at 60° north.
Where two zones meet
A twenty-kilometre baseline across a zone boundary is computed correctly in either zone — the two answers differ by nanometres — and taking one end from each returns 392 kilometres. Both zones are right and the seam between them is not a small error but a category one.
Where the worst point is
The largest scale error on a conformal map of a country is always on the frontier, never inside it, whatever the country's shape and whichever conformal projection was chosen. It is a theorem rather than a tendency, and it is the reason Chebyshev's criterion works.
The curvature of the Earth is not one number
An ellipsoid's Gaussian curvature varies by 1.35% from equator to pole, so the sphere is not developable onto the ellipsoid any more than the plane is onto the sphere. The spherical approximation is a projection with an irreducible cost, and the cost is 6,739 parts per million.
Choosing for a line, not a region
Every criterion in this subject integrates over an area. A pipeline, a railway or a coastal survey is a curve, and the projection an area criterion picks for it is not the one it should have — measurably, by a factor of five thousand.
A grid scaled to the ground is not a map
Construction sites work in coordinates where a tape reading equals a computed distance, which is achieved by multiplying the national grid by a constant. The result is exact at one point, degrades east–west forty times faster than north–south, and is not a map projection at all.
Grid north is not north
A grid has one north, parallel everywhere on the sheet by construction. The Earth has a different one at every point. The angle between them reaches two degrees at the edge of a UTM zone, and a straight line on the grid is not a straight line on the ground either.
What a tape measures
Four steps stand between an instrument reading and a coordinate, and their sizes are not in the order anybody expects. On a 77-kilometre line the slope reduction is 120 metres, the grid reduction 30, and a step nobody names — using the height above sea level where the height above the ellipsoid is wanted — is 583 millimetres.
A traverse must close
The check every survey specification leans on cannot see the error this whole field is about. Scale every leg of a closed figure by four hundred parts per million and it closes to the last bit of a double, while every dimension in it is wrong.
The ground is not the grid
A tape measure on a hillside has to be brought down to the ellipsoid and then out onto the map, and the two corrections have opposite signs. On a grid whose scale factor exceeds one there is exactly one elevation where they cancel — 2,551 metres, for a factor of 1.0004.
Two grids over the same ground
One point in the English Midlands is 406,788 east on the British grid and 167,478 east on UTM. Separating the difference properly leaves a surprise — only one of a grid's five parameters changes the map, and it is not any of the four that dominate the number.
Nearest is a question about the metric
Asked in the plane the data is stored in, a nearest-neighbour query returns a different site for 5 of 400 queries over Europe on Web Mercator — and 103 of 400 on the plate carrée, with the wrong answer up to 309 kilometres further away. Every disagreement is a contest closer than the region's own scale spread, and over a city there are none.
The tolerance decides the model
Every correction inverts to a distance — the line length at which it alone exceeds a stated tolerance. The ranking those distances produce is the same at a millimetre and at a decimetre, and it is different for two jobs at the same tolerance on different ground.
The places where a map is exactly right
Nine essays on this ladder say a map cannot be right everywhere. None asks where it IS right — and the answer is a curve, a pair of curves, or two isolated places, never a patch. Measured across fourteen projections the set's neighbourhood shrinks with an exponent of 0.48, 1.0 or 2.0, and the value the impossibility forbids is 0.
Zoom is a ladder
A tiling scheme exists only at integer zoom levels a factor of two apart, so a request for 1:25,000 is answered with 1:34,124 — 36 per cent coarser, and 86 per cent coarser in area. The mismatch runs from 1/√2 to √2 and repeats identically at every doubling, and nothing anywhere reports it, because the map that arrives is a perfectly good map of something.
An area on the grid is not an area on the ground
A grid's scale factor is a property of lengths and what a surveyor sells is an area. Squaring a departure doubles it, so a hectare drawn on the British grid at its central meridian has 10,008 square metres of ground under it — and correcting an area with the line factor instead of its square leaves half the error behind.
A coordinate is a number with a width
Every number on this site so far has been exact. A written coordinate is not: five decimal places of a degree is 1.112 metres of latitude everywhere and 1.112 metres of longitude only on the equator, falling to 0.097 at 85°. The same written precision carves the ground into a cell that is square in one place and eleven times longer than it is wide in another.
The best grid a country could have had
A national grid is a conformal map chosen for one region, so its whole design problem is one number: the spread of its scale factor. That number has a theoretical floor, this site can now compute it, and the adopted grid turns out to be either exactly optimal or half as good again — depending entirely on which box the country is declared to be.
The chain the satellite does not have
The four-step reduction from a tape reading to a grid coordinate exists because a tape does not know where it is. A satellite observation does, so the same distance can be got by projecting both ends and subtracting — and the two answers agreeing to a fifth of a millimetre is the first check the chain has ever had.
Not every distortion can be asked for
Six essays have written projections as conditions and asked how much freedom a condition leaves. The reverse question has never been put: a cartographer knows what distortion they want, so can they ask for it? For a conformal map the answer is a single equation, it is the Theorema Egregium in disguise, and asking for no distortion anywhere fails it by exactly the curvature of the sphere.
Sixty zones was a decision about one latitude
Twelve rungs price a grid, a zone, an origin and a reference, and every one of them works inside a single zone. The number of zones has never been asked about: six degrees meets its tolerance at the equator and is loose everywhere else, so a system spending the same tolerance evenly would use 51 zones at the equator and 8 at 82° — and UTM's worst error is 981 parts per million, not the 400 always quoted.
A slope is not a shape
Every map in this collection has carried geometry. An applied map far more often carries a field — elevation, pressure, a density — and the first thing anybody does with one is differentiate it. A gradient is a covector, it transforms by the inverse transpose of the Jacobian, and the ellipse that governs it is the indicatrix turned inside out.
A grid reference names a square
This ladder has priced everything about a grid except how a coordinate on it is written. A grid reference is truncated rather than rounded, so it names the south-west corner of a square rather than a point in it — and a population of references is displaced half a cell each way, which is a bias rather than scatter and does not average out.
The contour is right and the reading is wrong
There is exactly one thing about a field that no projection can get wrong: which points share a value. The contour lines on any two maps of the same field are the same set of points. Every quantity a reader takes off them — the spacing, the length, the area between two of them, the hypsometric curve — is not, and the two kinds of map get different ones wrong.
A projection defined by a table has an interpolation in it
One member of this library has no formula: Robinson set nineteen pairs of numbers by eye and the table is the definition. Three published interpolations of it draw graticules that differ by four parts in a thousand of the map's span and report angular deformations that differ by 8.4 degrees.
A chain of transformations does not close
Two datum transformations applied one after the other are not the sum of their fourteen published numbers. The gap is seven millimetres in Britain, doing the two in the other order moves the answer twenty, and the rotation matrix everybody prints is not a rotation.
One pair of numbers, a hundred and twenty places
A UTM coordinate is two numbers and a zone. Drop the zone and the numbers are still valid in each of the sixty; drop the hemisphere too and the pair names a hundred and twenty places. They form two rings at one latitude each, spaced exactly six degrees apart, and every one of them has the same grid convergence and the same scale factor — so no further geometric measurement can choose between them.
The seven parameters have their own uncertainty
Nine essays on this ladder print a datum transformation as seven exact numbers. Every published set is the output of a least-squares fit and arrives with standard errors as much a part of the result as the parameters — and pushing those widths through to the ground gives an ellipse, not a number, that is 68 mm across at the equator and 43 mm at 70°.
Four radii of the Earth
Ten rungs handle the ellipsoid with an auxiliary latitude. Every one of those constructions also needs a radius, and the radius that makes each property exact is a different number: the published 6371 km is right for an area to half a part per million and wrong for a meridian distance by 559 — while the radius a conformal map needs is not a constant at all, and spans 6,739.
Simplification does not commute with the projection
A pipeline either simplifies the geometry and then projects it, or projects it and then simplifies. Both orders are in use, neither is recorded, and given the same tolerance in ground metres they keep different vertices — 129 of them on the ground, 367 in degree space at 80°, and 459 on an equal-area page.
The sheet moved before it was measured
Nine rungs take control points off a map and assume the sheet they came from is the sheet the cartographer drew. Paper shrinks across its grain three times as fast as along it, and on a map whose grain runs along its own axis that shrinkage is EXACTLY a change of standard parallel — one per cent moves the recovered parallel by 0.57 degrees with the residual sitting at the solver's floor.
The strain a map adds to the ground's
A ground carried rigidly at forty millimetres a year deforms nothing, and read off a Mercator sheet at sixty degrees north it reports 10.9 nanostrain a year. Along a profile through a real boundary the invented part is 0.0 per cent of the answer where the zone is loud and 884 per cent where it is quiet.
The query a fast path actually answers
Fifteen rungs price what a stored coordinate means and none asks what it is searched with. No index answers "within two hundred kilometres of here"; an index answers "inside this rectangle of stored coordinates", and the rectangle is built by somebody's arithmetic — which at 55° north silently drops a tenth of the true answer on a conformal projection and three fifths of it on an equal-area one.
The curvature of a field is not the curvature of its picture
Whether a place is a spur or a hollow is the sign of a second derivative, and every automatic terrain classification is built on it. Read off an equal-area grid, that sign is wrong on 13.25 per cent of a sheet — and conformality reduces it to 0.30 per cent without removing it.
A tolerance in map units is not a tolerance
A snapping tolerance is a number, and the number is in whatever units the file is in. Five map units on Web Mercator is 4.97 metres of ground at the equator and 0.87 at eighty degrees — so a rule that merges two features three metres apart merges them everywhere below 52.8° north and refuses everywhere above it, in one pass, over one dataset, with nothing recording where the boundary is.
A symbol has a size on the page and an area on the ground
A proportional symbol is right as a total wherever it is drawn — a count is a count. Read against the region beneath it, which is how a reader forms a density, it is wrong by exactly the reciprocal of the areal factor: 0.083 at 73° north on Mercator, a factor of twelve, with the correction available as one multiplication that no atlas makes.
The lines where the bending vanishes
Every projection has a line printed in its margin — the standard parallel, where the scale is exactly one. It has a second special line nobody prints: the one along which a geodesic is drawn straight to second order. On Gall–Peters they are forty-five degrees apart, and the rule turns out to be tangency — a tangent construction puts both lines at its point of contact and a secant one moves only the first.
Named alongside it
The objects these essays reach for when they reach for this one.
ToleranceNational GridVerificationConformalityConventionZoneUTMMercatorPurposeTransverse MercatorCentral meridianPrincipal scale factors