Boundary — where it appears
Named by 20 essays across 8 fields — each of them below, with the objects they name alongside it.
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.
Total curvature and the scale rule
The impossibility has a size. A region covering a fraction of the sphere carries a fixed amount of curvature that any flat map must absorb, and for a circular region the least distortion any conformal projection can achieve is a closed form nobody can beat.
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.
Measuring curvature from inside
A flatlander with a ruler can find out that its world is round, by drawing a triangle and adding the angles. Gauss–Bonnet turns that into an exact statement, and the total curvature of a closed surface turns out to be a number obtained by counting.
A tile is drawn without its neighbours
Independence is what makes the scheme scale: one request touches one square of the world. It also means a feature crossing nine tiles is nine features, each labelled at its own centroid — up to 1,863 kilometres from the whole one — and the displacement is bounded by the size of the feature rather than the size of a tile, so it grows as the reader zooms out.
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.
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.
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.
A circle of a distance is not a circle
Eleven essays in this field have followed a route across a map. A range ring is not a route — it is the edge of a set — and drawing one exposes a failure the route essays cannot: the same ground distance comes out 4.17 times longer in one direction than another on a common projection, and 13.03 times at 70° north.
The antimeridian is a cut in the numbers
A twenty-degree box across 180° has a bounding box of 359.4°, a planar area seventeen times too large, and a midpoint 20,015 kilometres from where it belongs — which is the antipode, exactly. Moving the cut moves the failure and never removes it, because a circle cannot be numbered by an interval.
Nearest of many is a partition
One reach question with one source is a disc. With several sources it is a division of the whole surface, every place belonging to whichever source is nearest — and computing that division in the plane the data is stored in hands away between 0.75 and 22.16 per cent of the ground, in unbroken strips up to 1,591 kilometres across.
A corridor has a width the page cannot keep
Buffering a line is the most-run operation in spatial analysis and the corridor it produces is a reach set with two failures nobody separates: stroked on the page, one width covers 190 to 471 kilometres of ground along a single route; computed in closed form, the formula stops being the area at a width the route's own length fixes, and eventually claims more ground than the sphere has.
Inside is a claim about the edges
Whether a point is inside a polygon is not a property of the point and the polygon. It is a property of the plane the edges were understood to be straight in, and between two readings of the same file there is a band of disagreement — 29 per cent of one triangle's area, 4.2 million square kilometres, and the file does not say which reading it means.
The most compact shape depends on the paper
A geodesic disc attains the isoperimetric bound on a sphere — its score is one, exactly, at any radius. Score the same nine regions from their images on ten projections and four of them put something else on top, an oval and its own 45° rotation come out 4.7 per cent apart, and the projection that preserves the order is not the equal-area one.
The score is not stable at any scale
One boundary, read at eight resolutions from sixteen points to two thousand and forty-eight: the compactness score falls from 0.980 to 0.834 and is still falling. Changing the projection instead moves it by 0.69 per cent. The two decisions are made by the same person on the same afternoon and only one of them is ever reported.
The map depends on where it was cut
The previous rung solved the discrete conformal equations on a triangulated body and reported an areal spread of 3.07, then recorded that the number might belong to the patch, the boundary and the two pinned vertices rather than to the surface. It belongs to the pins: hold a different pair and it runs from 2.68 to 7.44, while the typical angular deformation does not move at all.
A polygon on a sphere has no outside
Seven essays have treated a stored ring as a boundary between inside and outside. A closed curve on a sphere divides it into two pieces and neither of them is the outside, so every polygon in every file depends on a convention that no coordinate carries — and the two conventions in common use disagree by a factor of fourteen on any ring that contains a pole.
Named alongside it
The objects these essays reach for when they reach for this one.
ToleranceScale factorNational GridVerificationZoneConventionPurposeReachRegional distortionUTMAreaClosed form