Concept

Buffer — where it appears

The set of points within a stated distance of a place, which is a disc on the sphere and is not a cell in any scheme. Answering a query about one costs more than the ratio of its area to a cell's, because the cells it overlaps have to be tested rather than merely counted.

Named by 6 essays across 3 fields — each of them below, with the objects they name alongside it.

"Within 0.01°" on the ground, at five latitudes. The same condition drawn at five latitudes, all at one scale. A degree of latitude is a fixed distance — 1106 metres here, varying by less than one per cent from equator to pole — while a degree of longitude collapses as cos φ, from 1113 metres to 289 at 75°. So the "circle" is an ellipse of 3.86:1 there, and it encloses 26 per cent of the ground the same condition covers on the equator. Even on the equator it is not round: M is smaller than N by the flattening, so the shape is 6694 parts per million shorter north–south than east–west.

A degree is not a unit of length

"Within 0.01 degrees" is a condition anybody can write and no instrument can measure. On the ground it is an ellipse — 1,106 metres north–south and 558 east–west at 60° — and even on the equator it is not a circle, because the meridian's radius of curvature is smaller than the parallel's by the flattening.

applied · Dataset

Everywhere within 200 km of the route from London to Tokyo. The shortest route between the two places, and the set of places within 200 kilometres of it, with both edges computed on the sphere and then projected. On the ground the set holds 3.949 million square kilometres — the band 3.823 and the two end caps 0.126, which between them make one disc of the corridor's own width. Drawn in Mollweide the corridor is visibly wider at one end than the other, and the ground it stands for is not.

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.

paths · Reach

A 6° query against a cube scheme, and the cells it fetches. The cells of a tangent-warped cube scheme at level 5, with the 29 cells a query of 6° radius touches shaded. The disc's own area is 16.84 cells; the count is 29, because every cell the disc's boundary crosses is fetched as well as every cell inside it. In Hilbert order those cells form six contiguous ranges of identifiers, which is six range scans, and the span from the lowest to the highest covers 91 cells against the 29 wanted. Drawn in Mollweide, with the mesh shown only near the query.

A query is a disc, and a disc is not a cell

Everything a cell system does is an address lookup except the one question anybody actually asks it: find everything within five kilometres of here. That is a disc, and the number of cells it fetches is not its area divided by a cell's — at the radii a query is really made at, it is three to seventeen times that.

applied · Cells

A bank that dries at one tide adds a bulge to the territorial sea. A stated coast 116 km long with five bays 5 km deep, and its twelve-nautical-mile limit measured from the low-water line at two tidal datums 1 m apart. A bank 10 km offshore has its crest 0.5 m below the higher datum, so it is under water at that datum and dries at the lower one, over a disc 1.0 km in radius. Dried and inside twelve miles of the coast, its low-water line becomes part of the baseline, and the limit bulges round it: 339 km² of sea. Everywhere else the two limits are drawn on top of one another, 11.1 km² apart over the whole coast. The coast is stated, not surveyed.

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.

practice · Boundary

Straight baselines across a fringe of islands. A stated coast 220 km long, a gentle undulation 10 km in amplitude, with 8 islands off it: six between 5 and 12 km offshore and two rocks at 30 and 26 km. The dashed line is the twelve-nautical-mile limit measured from the low-water line of the mainland and every island. The solid lines are the straight baselines enclosing the most water under one reading of the article — segments of at most 24 nautical miles departing at most 20° from the coast's general direction — 6 segments, the longest 44.4 km — and the solid curve is the limit measured from them. Tinted: 988 km² of water landward of the lines, now internal waters. Shaded: 68 km² of sea beyond the dashed limit, against a territorial sea of 8,122 km². Neither rock is reached. The coast is stated, not surveyed.

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.

practice · Boundary

The coast copied twelve miles out gives up the sea of every narrow bay. A stated coast 116 km long with five bays 5 km deep, 6, 10, 16, 24, 36 km wide. Dashed: the envelope of arcs, the line every point of which is twelve nautical miles from the nearest point of the coast. Solid: the coast itself moved twelve miles seaward, which dents the limit by the full 5 km over every bay. Tinted: the sea between them, 240 km² in all, 9% of a straight coast's territorial sea. Over the 6 km bay the copy lies 4.8 km inside the envelope; over the 36 km bay the two meet. The coast is stated, not surveyed.

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.

practice · Boundary

Named alongside it

The objects these essays reach for when they reach for this one.

Boundary Convention Tolerance Verification Area Baseline Closed form Radius of curvature Address Aggregation Anisotropy Cell

All concepts