Cartographic Projection
every map is wrong, and by exactly how much
Figure

An 8° query against a cube scheme, and the cells it fetches

Drawn here at the parameters it defaults to, with every essay that calls it.
An 8° query against a cube scheme, and the cells it fetches. The cells of a tangent-warped cube scheme at level 3, with the 4 cells a query of 8° radius touches shaded. The disc's own area is 1.87 cells; the count is 4, because every cell the disc's boundary crosses is fetched as well as every cell inside it. In Hilbert order those cells form two contiguous ranges of identifiers, which is two range scans, and the span from the lowest to the highest covers 6 cells against the 4 wanted. Drawn in Mollweide, with the mesh shown only near the query.

The cells of a tangent-warped cube scheme at level 3, with the 4 cells a query of 8° radius touches shaded. The disc's own area is 1.87 cells; the count is 4, because every cell the disc's boundary crosses is fetched as well as every cell inside it. In Hilbert order those cells form two contiguous ranges of identifiers, which is two range scans, and the span from the lowest to the highest covers 6 cells against the 4 wanted. Drawn in Mollweide, with the mesh shown only near the query.

It is drawn by cell-figure with show: "cell-query" — one member of a family of 5 figures that share a generator, so the drawing above is what that generator returns when it is asked for this one and given nothing else.

3 essays call it. Every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

Where it is called

Changing this changes every one of these figures.

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. What a machine does with it

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.

 Hilbert order on one face, as a curve. The order in which Hilbert numbering visits the 64 cells of one cube face at level 3. The line never leaves a cell without entering one that shares an edge with it — that is what makes it a space-filling curve, and it is why two cells with nearby identifiers are usually near each other on the ground. What a machine does with it

The address is a curve through the sphere

A database does not fetch a set of cells, it reads ranges of identifiers — so the cost of a query is how many runs its cells form, not how many cells it needs. Hilbert order wins that measurement and loses the one usually quoted for it: its neighbouring cells are further apart in identifier than row-major's, on average and at worst.

 The piece two schemes share, clipped rather than assumed. A cell of a gnomonic cube, whose four edges are great-circle arcs because a straight line on a gnomonic face is one, against a cell of a longitude–latitude grid, whose north and south edges are parallels and are not. Their overlap is neither a rectangle nor a spherical polygon of any standard kind, and it is 5965687 km² of the cube cell's 5965687 km² — 100.0 per cent. Computing it needs the arc of one boundary intersected with the plane of the other, which is three equations and two roots, and it is exact. What a machine does with it

Cells that are rectangles in no coordinate

The previous rung measured what moving a field between two cell schemes costs, and did it between two schemes whose cells are longitude–latitude rectangles — which is what made every overlap a rectangle with a closed-form area. The schemes anybody actually argues about have cells that are rectangles in no coordinate, and their overlaps have to be clipped.

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