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

Principal direction — where it appears

The direction at a point along which a projection's scale factor is greatest, and the perpendicular one along which it is least. It exists wherever the map is not conformal and is undefined where it is, since a circle has no long axis.

Named by 8 essays across one field — each of them below, with the objects they name alongside it.

Which way Sinusoidal stretches the ground. The major axis of Tissot's indicatrix at 180 points, drawn as a stroke rather than an ellipse because the argument is the orientation. Longer strokes mark more elongated indicatrices. The direction departs from the graticule by up to 44.4°, so the scale "along the meridian" is not the scale along the direction that is actually stretched most.

Distortion has a direction

Tissot's indicatrix is an ellipse, an ellipse has an orientation, and the orientation is never reported. On most projections it is not along the meridian, on a conformal one it does not exist at all, and both facts are computable from the same four derivatives as everything else.

distortion · Tissot One instrument, one covariance, drawn where Mollweide puts it. The same measurement is made at every point: 5 m east by 5 m north, uncorrelated, which on the ground is a circle. Each ellipse is that covariance pushed through the projection's own Jacobian and drawn 26,000 times life size. On Mollweide the axis ratio reaches 4.15, so an instrument that is equally good in every direction is drawn as though it were not.

An error ellipse is an indicatrix

A positional covariance pushed through a projection is the same matrix sandwich that produces Tissot's indicatrix, so the error ellipse drawn on a map and the distortion ellipse drawn beside it are the same ellipse. A five-metre circular accuracy is drawn at an axis ratio of 3.04 on one common projection — and the same projection draws a genuinely lopsided 304 by 100 metre error as a perfect circle.

distortion · Precision Tissot's ellipse and the one that governs gradients, at 20°E 48°N. The solid ellipse is the image of a small circle — Tissot's indicatrix, semi-axes a and b. The dashed one is the image of a unit gradient, whose semi-axes are 1/b and 1/a because a gradient transforms by the inverse transpose of the Jacobian rather than by the Jacobian. Its long axis therefore lies where the indicatrix's short one does. On Mercator both are circles, so a gradient's direction survives; on Lambert cylindrical the two ellipses are the same shape turned through a right angle, so the worst direction for a gradient is the best direction for a shape.

A slope is not a shape

Every map in this collection has carried geometry. An applied map far more often carries a field — elevation, pressure, a density — and the first thing anybody does with one is differentiate it. A gradient is a covector, it transforms by the inverse transpose of the Jacobian, and the ellipse that governs it is the indicatrix turned inside out.

distortion · Gradient Steepest descent on Lambert cylindrical, computed on the ground and on the page. Fifteen routes, each started at the same place twice. The solid line follows the true direction of steepest descent on the sphere; the dashed line follows the direction read off the page at every step, which is what an analysis of a projected grid does. Both take the same length of step on the ground, so the only difference between them is the direction. They part by up to 1517.8 kilometres, against a bearing error of 45.6°. The faint lines are contours of the field, which is seven caps at stated centres and widths.

Water runs downhill on the ground, not on the page

A drainage network is the set of steepest-descent trajectories of a field, so it is built entirely out of directions. A conformal map preserves those directions exactly and therefore preserves the whole network; an equal-area map does not, and sends a route up to 892 kilometres away from where the water actually goes.

distortion · Gradient Four averages of one region's indicatrices — Robinson over the whole sphere. The faint ellipses are the indicatrix at ninety sampled places; the four drawn over them are four candidate averages of exactly that set. The arithmetic mean of the tensors gives semi-axes 1.1621 and 1.0182; the log-Euclidean mean gives 1.0053 and 0.9608; the Karcher mean gives 1.0042 and 0.9619; and taking the mean of a and the mean of b separately — which is what a published average distortion usually is — gives 1.2335 and 0.8200. Their maximum angular deformations are 7.57°, 2.60°, 2.47°, 23.23°, against a mean of the pointwise deformations of 20.67°. Five numbers, one set of ellipses.

An average of ellipses is not an ellipse

Rung three integrates distortion over a region and finds the weighting is somebody's opinion. It never asked what was being averaged. An indicatrix is a positive-definite matrix, matrices form a cone rather than a vector space, and the arithmetic average of an equal-area map's indicatrices comes back inflating area by up to 11 per cent.

distortion · Tissot 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 points a projection cannot move. The critical points of a stated field, found twice: once on the sphere from its own gradient, and once in Lambert cylindrical's page coordinates from the page's own numbers, with nothing shared between the two searches. seven points, 3 maxima, 2 saddles and 2 minima, and the two sets agree in position to 2.3e-7 degrees and in type at every one. The page's Jacobian is invertible wherever the map is a map, so it sends a zero gradient to a zero gradient and cannot move a critical point anywhere.

What the page cannot move

Six rungs measure what a page does to a field's readings and every one of them moves. The critical points do not: found on the sphere and found again in a projection's own page coordinates with nothing shared between the searches, they agree to 10⁻⁷ degrees and in type at every point, on every projection. What the page does move is their shape, by a factor bounded exactly by the indicatrix's axis ratio squared.

distortion · Gradient What a reprojection does, drawn where it does it. The angular deformation of the map that takes Sinusoidal's page to Gall–Peters's — which is what a reprojection is — shaded over the sphere and drawn on Mollweide so that equal ground areas are equal page areas. Both parents are equal-area, so the composite preserves area exactly; its mean angular deformation is 55.5° against 38.9° and 32.2° for the two maps separately. Composing two maps did not average their shape errors; it made a larger one.

Two indicatrices do not make a third

Reprojecting is composing, and the composite's indicatrix is not a function of its parents'. Mollweide followed by Hammer is gentler than either map over eighty per cent of the sphere; the sinusoidal followed by Gall–Peters is worse than both everywhere. What separates them is one angle, and it is the number rung ten showed the ellipse does not carry.

distortion · Tissot

Named alongside it

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

ConformalityAnisotropyEqual-areaTissot's indicatrixAngular deformationInvariantJacobianGradientIndicatrixSingular valuesAreal factorDuality

All concepts