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

Page frame — where it appears

The coordinate system of the sheet itself, in which a raster's rows and columns run and in which every neighbourhood operator computes. A quantity computed there is a page quantity, and turning it into a ground quantity needs the projection the operator does not have.

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

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 A legend saying "illuminated from 315°" is true at one longitude. A hillshade's azimuth is measured from the top of the sheet, because the shading is computed on the projected raster. The top of the sheet is grid north, so the compass bearing the light comes from is the declared azimuth plus the meridian convergence, and that varies across the sheet. On a conic it swings by 55.3 degrees over eighty degrees of longitude. On a cylindrical projection in its normal aspect it does not swing at all, which is the flat line — the only case the legend is right everywhere.

The light comes from a page direction

A hillshade's illumination azimuth is declared from the top of the sheet, and the top of the sheet is grid north. On a conic the light therefore swings 52.6° across eighty degrees of longitude, and 72.8 per cent of the sheet is shaded differently from what the legend claims.

distortion · Gradient Conformality helps and does not save it. The proportion of a stated terrain whose plan curvature changes sign when it is read off a grid in each projection. A conformal map turns every direction through the same angle, so the contour and the slope line stay perpendicular and the sign ought to survive — and it mostly does, at 0.30 per cent against 13.3. It is not zero, and the term that flips it is the gradient of the scale factor: the curvature of a curve under a conformal map is (κ − ∂ₙ log λ)/λ, and Mercator's λ has a gradient.

The curvature of a field is not the curvature of its picture

Whether a place is a spur or a hollow is the sign of a second derivative, and every automatic terrain classification is built on it. Read off an equal-area grid, that sign is wrong on 13.25 per cent of a sheet — and conformality reduces it to 0.30 per cent without removing it.

distortion · Gradient

Named alongside it

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

RasterAspectClippingConformalityConicDerived productDiscretisationFlexionGradientGrid northHillshadeHollow

All concepts