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

Pixel — where it appears

One cell of a screen raster, which stands for a patch of ground whose size depends on the zoom and the latitude. It is a place with an extent rather than a point, so every value it carries is an average over an area rather than a sample at a position.

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

Equal bands of screen, unequal bands of ground. A section through a camera pitched 60° from straight down, at the height that makes an untilted view exactly the flat map — 1152 pixels above the plane for a 768-pixel screen and a 36.87° field of view. The rays are drawn at equal spacings up the screen and the blocks below them are the ground each of those equal bands covers: 1.75 plane units per pixel at the bottom of the frame and 16.88 at the top, a factor of 9.7. Nothing about this is a projection of the sphere; it is a photograph of a map, and the map underneath it is still Web Mercator.

A tilted view has no zoom level

Every essay about the screen so far assumes the map lies flat on it, which was true until about 2015. Pitch the camera sixty degrees and one frame asks for 3.6 zoom levels at once, a square tile covers ground four and a half times deeper than it is wide, and the pyramid has one integer per tile to answer with.

applied · Screen One of these three lines does not slope. Three lengths at each zoom, for a world coordinate near the antimeridian. The pixel halves with every level, as it must. The spacing between representable double-precision values is far below it and halves with it. The spacing between representable SINGLE-precision values does not move at all — 2.0 metres at every zoom, because it is a property of the size of the number and the world does not get smaller. The two cross at zoom 17, and past it a vertex snaps to a lattice coarser than the pixels it is drawn into.

The renderer runs out of numbers before the zoom does

Eleven essays price the pyramid in exact arithmetic. The pipeline that draws it carries single precision, where a world coordinate near the antimeridian quantises to two metres — at every zoom, because the number does not get smaller when the pixel does. At zoom 22 that is fifty-four pixels.

applied · Screen The same labels, placed once and placed per tile. 140 stated label boxes on a 1024-pixel page cut into 256-pixel tiles. The pale boxes are placed the same way by both rules. The dark ones are placed by the tiled renderer and suppressed by the global one — labels that should have lost a collision with something in the next tile and did not, because the tile that drew them could not see it. nine of them, against 104 labels the global rule keeps.

A label belongs to no tile

Nine rungs price the tile as a piece of geometry. A label is not geometry — it is a page object placed by collision against other page objects, and collision is a global relation while a tile is rendered alone. Cut a page into tiles and 10.3 per cent of the labels are placed differently; the buffer that closes the gap is half a tile at moderate density and a whole one when it is crowded.

applied · Screen

Named alongside it

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

QuantisationTile pyramidVector tileWeb MercatorZoom levelAnisotropyCameraClippingDiscretisationFloating-pointGround resolutionHorizon

All concepts