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

Raster — where it appears

A grid of sampled values, each of which is a place with a size rather than a point, and which cannot be reprojected without inventing values. Reprojecting one invents values, because the output cells do not sit on the input cells and something has to be read between them.

Named by 10 essays across 3 fields — each of them below, with the objects they name alongside it.

A raster warped to Lambert azimuthal equal-area and back, nearest against bilinear. The left panel is the field the raster carries — a smooth analytic function, so that the error of an interpolation is the interpolation's error and not a photograph's history. The other panels are what is left after warping into Lambert azimuthal equal-area and back to Equirectangular, shown as the difference from the original at six times the contrast. Nothing moved: the coordinates go through the maps exactly. What is lost is that a target pixel's centre does not fall on a source pixel's centre, so a value has to be invented for it. Bilinear is closer to the field — RMS 0.0022 against 0.0212 — and has given up 0.63 per cent of its variance to get there. The panels are drawn at 48 by 32 cells; the measurement is made at the same resolution.

Reprojecting a raster invents values

Moving a picture from one projection to another moves no coordinate — the maps are exact both ways. What is lost is that a target cell's centre does not land on a source cell's centre, so a value has to be made up for it, and the making-up has an order of convergence: 1.00 for nearest, 1.98 for bilinear, 2.93 for a cubic, measured by refining the grid.

applied · Dataset The same kernels, across an edge. Each kernel measured over the same five rasters as the smooth measurement, on a field that carries a step across a tilted line instead of a smooth function. The solid lines are the edge and the faint ones the smooth field. The orders on the smooth field are nearest 1.00, bilinear 1.98, cubic 2.93; across the edge they are nearest 0.78, bilinear 0.58, cubic 0.60 — within a factor of 1.33 of one another, and the fastest of them belongs to nearest-neighbour, which does no interpolating at all. The ranking a smooth field establishes does not survive a discontinuity, and a real raster is mostly edges.

An edge has no order of convergence

On a smooth field the three resampling kernels converge at orders 1, 2 and 3 and the choice is obvious. Across a discontinuity they converge at 0.78, 0.58 and 0.60 — within a factor of 1.4 of each other, in an order that puts nearest-neighbour first, and a real raster is mostly edges.

applied · Dataset The convergence order across an edge, against the edge's own orientation. Each curve is one kernel, fitted the same way as every other convergence order on this site: the root-mean-square error against the grid spacing, in logs, over five refinements. At 27° they read 0.78, 0.58, 0.60, which is the measurement already published here — and 27° is one point. Turn the edge onto a parallel and the curves collapse, and the nearest-neighbour one goes negative, which is the fit's way of saying the error is not falling at all. A single number for "the order across an edge" is a number about the edge that was measured.

One edge is not an edge

The three resampling kernels were measured across a discontinuity and came out at 0.78, 0.58 and 0.60 — one straight edge at 27° to the graticule. Across thirteen edges the same kernels span 0.19 to 0.87, the ranking between them reverses, and for an edge lying along a parallel the error does not fall with refinement at all.

applied · Dataset Give the edge a width and the kernels get their orders back. Every edge this collection has resampled across has been exactly discontinuous, which is not what a sensor produces: a footprint, an atmosphere and a lens all smooth a boundary over a cell or two before anything is resampled. Convolving the edge with a Gaussian of stated width and refitting gives 1.23, 1.97 and 3.60 at one degree of blur, against 0.78, 0.58 and 0.60 with no blur at all. The blur is held fixed in degrees while the grid refines, which is what happens to a real sensor's data as its resolution improves.

A real edge has a width

Thirteen edges were measured and every one of them was exactly discontinuous, which no sensor has ever produced. Convolving them with a point-spread function of one degree — a cell or two — takes the three kernels from 0.78, 0.58 and 0.60 back to 1.23, 1.97 and 3.60, and takes the edge along a parallel, which converged at −1.49, up to 1.92 for bilinear and 3.73 for cubic.

applied · Dataset 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 Eighteen ways to close an archipelago within a hundred miles. A stated archipelago about 580 km across: three main islands near the centre (A, B and C, 23,576 km² of land between them) and thirteen smaller islands round them, 26,970 km² of land in all. The dot on each island is its outermost point, where a state drawing the widest lawful system would put its basepoint. Solid: the lawful system enclosing the most water, 186,617 km² at a water-to-land ratio of 6.96 to one, 12 segments, the longest 86.6 nautical miles. Dashed: the lawful system enclosing the least, 159,032 km², through the main islands A, B and C. Dotted: the two segments that would take in the outlier P to the south-east, 122.3 nautical miles and 100.9 nautical miles — both over the hundred-mile limit. The archipelago is stated, not surveyed.

Written in numbers, the archipelago rule still leaves eighteen

The article on archipelagic baselines gives its tolerances as numbers: no segment over a hundred nautical miles except three per cent of them, which may reach 125, and water between one and nine times the land. On a stated archipelago 8,799 closed systems keep the main islands inside, the ratio passes all of them and the length test eighteen, and the eighteen still differ by 27,585 square kilometres. Thin the land and the ratio does the deciding; the lower bound decides nothing; and the three-per-cent exception can be used only by a system padded to thirty-four segments.

practice · Boundary Two lawful systems, one sea: the lines move water between regimes and the outer limit does not move. The stated archipelago with the lawful system enclosing the most water (solid, 186,617 km² inside) and the one enclosing the least (dashed, 159,032 km²). Round each, the limit two hundred nautical miles from its baselines and from every island's coast: the two outer limits lie on top of each other everywhere but a few kilometres off the widest gaps. A straight coast stands 700 km east of the centre; the line equidistant between it and the archipelago is drawn for both systems and they coincide. Inside plus outside, the two systems account for 1,344,857 km² and 1,343,864 km² of sea.

The archipelago rule moves water between regimes, not between states

The eighteen baseline systems the archipelago article allows on a stated archipelago differ by 27,585 square kilometres of water inside the lines. Measured outward, as the Convention measures every zone, the difference almost vanishes: whatever a system takes inside it gives up from the zone outside, and water inside plus zone outside differs by 993 square kilometres, seven hundredths of a per cent. Against a straight neighbouring coast on any side, the equidistance line does not move at all, because a baseline's nearest point to a straight coast is always an island. Only a neighbour facing the middle of a gap feels the choice — 2,112 square kilometres at thirty kilometres off, 688 at 250.

practice · Boundary 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 One map, two samples, two answers. The same Miller cylindrical projection carrying two sets of sample points of the same size. On the left the points are uniform over the SPHERE — equal ground area between them, which is what every mean in this collection integrates against. On the right they are uniform over the PAGE, which is what a raster, a pixel loop or any figure that walks its own canvas produces. The right-hand set crowds where the map stretches, and the mean angular deformation it returns is 18.0° against the left-hand set's 7.2°.

The sample was drawn on the page

Every mean in this collection integrates over the sphere, because that is where the ground is. A raster, a pixel loop and any figure that walks its own canvas integrate over the page instead, and the difference is exactly the covariance between the quantity being measured and the map's own area distortion — 7.2° of mean angular deformation on Miller becoming 18.0°.

distortion · Sampling

Named alongside it

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

ResamplingToleranceVerificationConvergence orderBaselineBoundaryConventionDiscontinuityWarpAliasingAreaClosed form

All concepts