Concept

Bounding box — where it appears

The smallest axis-aligned rectangle containing a feature, in whatever coordinates the feature is stored in. It is a cheap proxy for a region's shape and a poor one: two rules of thumb for choosing a projection both read one, and both collapse on regions whose long axis is oblique.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

A 20° shape across the antimeridian, in the space where the numbers live. Longitude runs across the page from −180° to 180°, which is where the failure is: the shape is one rectangle on the ground and two pieces in the numbering, and every operation that treats longitude as a real number sees the two. The bounding box comes out 359° wide instead of 20°, the planar area comes out 17 times too large because the shoelace encloses the complement, and the midpoint of a segment from one edge to the other lands 20015 kilometres away — the antipode of where it belongs. The true area, from the closed form, is 4,920,667 square kilometres.

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.

applied · Dataset

Two rules, three populations. The taught rule keys on latitude; the replacement keys on shape. On the thirty regions the replacement was read from it scores 83 per cent against the taught rule's 63. On forty-five different regions built the same way it scores 91 — higher, not lower, so the generalisation the shortfall doubted is real. On the seven named regions this collection actually uses, both rules score 29 per cent, and following either costs a mean factor of 14.6. The third bar of each group puts the taught rule's polar clause back into the shape rule, which is what the out-of-sample failures ask for: it repairs four of them, breaks two that were right, and reaches 43 of 45.

The rule scored out of sample

A replacement rule was read off thirty regions and scored on the same thirty, and this collection recorded that as not being evidence about any other thirty. It is: on forty-five different regions the rule scores 91 per cent against the 83 it managed at home. What it cannot do is the seven regions the collection actually uses, where both it and the rule it replaced name the winner twice out of seven and cost a mean factor of 14.6.

wrong · Audit

The query, the box it becomes, and the 14 per cent it loses. A query for everything within 400 kilometres of a place at 55° north, drawn in the stored coordinates of Web Mercator. The ring is the true answer's edge; the rectangle is the box the index is given, sized from the scale factor at the query point. The arcs outside the rectangle are true neighbours the index never returns — 13.6 per cent of the rim, reaching 5.0 per cent of the box's own half-width beyond it — and nothing downstream can tell they are missing.

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.

applied · Dataset

Named alongside it

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

Purpose Verification Antimeridian Audit Azimuthal Boundary Branch cut Closed form Conformality Conic Convention Coordinate system

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