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

Orientation — where it appears

Which way a feature lies relative to a grid or a graticule, which decides how a resampling kernel behaves across it and is invisible in any average. A resampling kernel's convergence depends on it, and an aspect search's objective is stationary in the rotation that a symmetric region is unchanged by.

Named by 6 essays across 3 fields — each of them below, with the objects they name alongside it.

Every mixture of Equirectangular at 50.46° and Aitoff. Airy's criterion over the whole sphere, for every weighted average of the Winkel tripel's own pair. Each mixture is normalised to its own best constant scale first, so the comparison is about shape rather than size. The curve has its minimum at w = 0.45, where the criterion is 0.3037 against 0.4114 and 0.4290 at the two ends. That is a projection nobody constructed beating both projections somebody did, by 26 per cent.

The average of two projections

The Winkel tripel is literally the arithmetic mean of two other projections, and this site's implementation of it agrees with that mean to the last bit. Averaging beats both ingredients by 26 per cent — and it preserves conformality exactly, destroys equal-area completely, and can turn eighteen per cent of the world inside out without either distortion measure saying so.

choosing · Choosing 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 A ring round the pole at 80°, and the two pieces it makes. the boundary of a small polar cap — and of everything else. A closed curve divides a sphere into two pieces and neither of them is the outside: one is 4 thousand square kilometres and the other is 506 thousand, a ratio of 130.6 to one, and the coordinates are the same either way. The two colours are the two pieces, sampled at points rather than shaded, because shading one of them would already be the decision this figure is about.

A polygon on a sphere has no outside

Seven essays have treated a stored ring as a boundary between inside and outside. A closed curve on a sphere divides it into two pieces and neither of them is the outside, so every polygon in every file depends on a convention that no coordinate carries — and the two conventions in common use disagree by a factor of fourteen on any ring that contains a pole.

applied · Dataset How aligned a region's ellipses are, for every projection and every region. The resultant length of the doubled indicatrix orientations, over 12 projections and 9 regions. One means every ellipse in the region points the same way; zero means they are spread evenly and cancel. Three cylindrical rows are 1.00 throughout, and everything else varies down the row and across it — which is the answer to the question the number was first stated without: alignment is a property of the pair. Mercator's row is the exception that is not a measurement, because a conformal projection's indicatrix is a circle and a circle has no orientation.

Whether the ellipses point the same way

The previous rung found that alignment decides whether the average of a region's deformations is above or below the deformation of its average, and stated it at ten projections over one region. Swept over a hundred and eight pairs, the answer is that alignment belongs to the projection 28 per cent, to the region 38, and to neither 33 — and two rows of the table turn out not to be measurements at all.

distortion · Tissot The same tiling, turned. The cube's eight vertices, ringed, and eight cities, with the aspect of the cell each city falls in written beside it. The two panels are the same tiling: the same cells, the same areas, the same shapes, in the same numbers. Only where they sit has changed, and the eight cities get cells whose areas spread by 1.26 in one and 1.39 in the other. Drawn on Mollweide, in which equal ground areas are equal page areas.

The orientation is a policy

A polyhedral cell system has three free angles nobody scores. They cannot improve it: rotating the solid rotates every cell rigidly, so the distribution of cell areas is identical for every orientation there is. What they decide is who stands on the bad cells — and the eight cities measured here get a spread of cell area of 1.00 under the best turn and 1.50 under the worst.

applied · Cells

Named alongside it

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

AliasingAreaConformalityConventionConvergence orderDiscontinuityInterpolation kernelRasterRefinementResamplingSampling latticeTest field

All concepts