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

A raster warped to Lambert azimuthal equal-area and back, nearest against bilinear

Drawn here at the parameters it defaults to, with every essay that calls 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.

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.

It is drawn by operation-figure with show: "warp-loss" — one member of a family of 14 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.

5 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.

What 4 parts per million of network strain leaves behind. Each arrow is where the best-fitting seven-parameter transformation leaves a marker, over 36 markers laid out as a grid across OSGB36's ground. The RMS residual is 1.64 metres and the worst is 3.00 metres. Seven parameters span the constant and linear parts of a displacement field; this one is quadratic, so no choice of the seven can reach it. The arrows are drawn 27× life size. What the numbers refer to

Where a fit leaves residuals

Seven parameters can carry a rigid motion and a size exactly. A triangulation network is neither, so the best possible transformation between two datums leaves metres on the table — in a pattern, not as noise — and which seven parameters come out depends on where the markers were.

 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. What a machine does with it

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.

 The circle a reprojection is about to turn into an ellipse. A circle drawn on the Mollweide plane, and its image under the map that carries that plane to Mercator — normalised so each pair has the same area, because the whole map can be rescaled and the shape is what is being shown. The dashed circle is what an undistorted reprojection would leave. This is Tissot's construction with a plane in place of the sphere, and it is the right picture for a reprojection because both ends are pictures. Worst angular deformation over the sampled points: 81.2°. Drawn on the source plane, in Mollweide. Measuring distortion

A projection between two projections

Every distortion measured on this site so far compares a map with the sphere. The operation a machine actually performs compares a map with another map — and that map has its own two principal scales, its own areal factor and its own angular deformation, none of which is the difference of the two it was built from.

 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. What a machine does with it

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.

 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. What a machine does with it

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.

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