Boundary — where it appears
Named by 34 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 anyone 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 shortest route between two coasts
A shortest path is usually asked for between two points. Between two regions the answer is attained at a pair of boundary points, and which pair is not the pair any page makes look nearest — a cylindrical map picks a pair 2,135 kilometres from the right one across the Atlantic, and 1,716 kilometres wrong between Chile and New Zealand, where its route is 22.5 per cent long.
A drawn reach set stops at the river
A polygon drawn through a fan of bearings round a reach set is short by a chord deficit that falls as the square of the count — on ground with nothing in the way. Put a river three kilometres off, crossable at bridges, and the polygon misses 30 per cent of the reachable ground at every count from forty-eight bearings to a thousand, and a convergence check passes on it.
One sentence, and the ground between its readings
Every land boundary in the world is defined by a sentence, and a sentence naming two monuments does not name a curve. "A straight line" between the Rio Grande and the Colorado admits at least three answers 7.53 kilometres apart at their worst; "the forty-ninth parallel" admits four, 96 kilometres apart, with 130,972 square kilometres between the extremes.
The map depends on where it was cut
Solving the discrete conformal equations on a triangulated body gave an areal spread of 3.07, with the number possibly belonging 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.
The line a commission can actually run
A boundary commission cannot run a parallel of latitude. It can sight a straight line between monuments, and a straight line between two points of equal latitude passes poleward of the parallel — by s² tan φ / 8R, which at a mile of spacing is fifty-eight millimetres and at a hundred kilometres is two hundred and twenty-six metres. The described line and the marked line are different curves, and the marked one governs.
A meridian boundary moves when its datum does
The 141st meridian is the one boundary description in this collection with no geometric ambiguity in it: every reading of it is the same curve, exactly. It has a different one. A longitude refers to a datum, the 1825 convention named none, and the line of longitude exactly 141° west sits 129.0 metres apart on NAD27 and WGS84 — a ribbon 1,039 kilometres long and 134 square kilometres in area.
An equidistance line belongs to a surface
A maritime boundary is very often defined as the line equidistant from two coasts — a description with no coordinate in it and no curve to choose between. It has a third ambiguity: equidistant measured how. On the ellipsoid, on the sphere, and with a ruler on three different charts, the same four basepoints give lines up to 39.6 kilometres apart and 2,816 square kilometres of seabed between them.
A tripoint defined three times
A tripoint is very often not a coordinate in any treaty. It is a description — the point where the boundary between A and B meets the boundary between B and C — and each of those boundaries is itself a description. So the point is defined three times, once by each pair, and under one convention throughout the three definitions agree to half a micrometre. Under three they enclose 6.69 square kilometres.
A river boundary goes where the river goes, or stays where it was
A boundary that follows a river moves when the river moves, and the law of every such boundary turns on one word: whether the river moved gradually or suddenly. Over one meander's life, gradual movement changes the country of about twenty-eight square kilometres of land a loop. A single flood that cuts the loop off strands twenty-six — nearly the same land, arriving as one parcel on the wrong side of the river.
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.
The highest summits stay where they are, and the watershed does not
A treaty that puts a boundary on the highest crests which divide the waters names one line where the water divides on the summits and two where a river has cut back through the range. On a stated range the two part by up to eighteen kilometres. They also differ in kind: an error of ten metres in every height moves the summits three metres and the watershed on the plateau more than a kilometre.
The sea is measured from a tide, and the headlands decide how much
A territorial sea is measured from the low-water line, and the low-water line is where a chart's tidal datum meets the shore — so a datum a metre lower moves it twenty metres at a headland and a kilometre across a tidal flat. On a stated coast the twelve-mile limit barely notices the flats: fifty-six square kilometres of shore are uncovered and eleven of sea are gained. A bank that dries at the lower datum adds three hundred and thirty-nine in one step.
A straight baseline moves more water inward than sea outward
The seventh article of the law of the sea lets a coast fringed with islands measure its territorial sea from straight lines joining chosen points, and gives no length, no angle and no scale for the choice. On a stated coast the lines a common reading permits add sixty-eight square kilometres of sea at the outer edge and turn nine hundred and eighty-eight into internal waters. Drop the test of direction and the outer rocks come into reach: eighteen hundred added, forty-six hundred enclosed.
Written in numbers, the archipelago rule still leaves eighteen
The article on archipelagic baselines gives its tolerances as numbers: no segment over a hundred nautical miles except three per cent of them, which may reach 125, and water between one and nine times the land. On a stated archipelago 8,799 closed systems keep the main islands inside, the ratio passes all of them and the length test eighteen, and the eighteen still differ by 27,585 square kilometres. Thin the land and the ratio does the deciding; the lower bound decides nothing; and the three-per-cent exception can be used only by a system padded to thirty-four segments.
The archipelago rule moves water between regimes, not between states
The eighteen baseline systems the archipelago article allows on a stated archipelago differ by 27,585 square kilometres of water inside the lines. Measured outward, as the Convention measures every zone, the difference almost vanishes: whatever a system takes inside it gives up from the zone outside, and water inside plus zone outside differs by 993 square kilometres, seven hundredths of a per cent. Against a straight neighbouring coast on any side, the equidistance line does not move at all, because a baseline's nearest point to a straight coast is always an island. Only a neighbour facing the middle of a gap feels the choice — 2,112 square kilometres at thirty kilometres off, 688 at 250.
Copied out to sea, a coast keeps its length and loses its bays
Before 1930 a territorial sea was drawn by following the coast in all its sinuosities, and the phrase fits two constructions. Moved twelve miles seaward, a stated coast of five bays gives up 240 square kilometres against the envelope of arcs, most of it over the narrowest bay; moved along its own normals, it ties a loop over every bay. On a fractal coast charted at finer detail the copy grows by a quarter at every level, to 480 kilometres, while the envelope settles at 133.0.
Named alongside it
The objects these essays reach for when they reach for this one.
ToleranceVerificationConventionAreaPurposeClosed formScale factorGeodesicGreat circleNational GridReachZone