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

Series truncation — where it appears

Stopping a power series after a stated number of terms and accepting the error that leaves. A national grid's forward and inverse maps are separately truncated series, so composing them does not return the point and the residual is not either half's.

Named by 16 essays across 6 fields — each of them below, with the objects they name alongside it.

Transverse Mercator. The graticule of the Transverse Mercator projection at 30° of longitude and 15° of latitude. the only projection on which a constant compass bearing is a straight line. It is conformal.

Transverse Mercator and the series that computes it

The projection most of the world's survey data lives in has no closed form. What every national grid actually computes is a truncated power series in the flattening, and how far it can be trusted is an engineering parameter rather than a property of the projection.

families · Ellipsoid Molodensky against the exact route, NAD27. The distance on the ground between where each shortcut puts the transformed point and where the exact Cartesian route puts it, at 0 metres of ellipsoidal height, on a logarithmic scale. The full formulae stay within 0.8 centimetres across every latitude drawn; the abridged form, which replaces two ellipsoid-difference coefficients with one combined term, is worst at 40 centimetres — a factor of 48. The abridged curve dips near 45°, where the combined coefficient happens to equal the pair it replaces.

Molodensky's shortcut

The exact way to shift a datum needs an iteration nobody wanted to run on a 1950s machine, so a direct formula was derived instead. It lands within centimetres — and the abridged version everybody quotes is a different formula, worse by a factor of fifty.

datums · Datum The image of a circle on Mercator, against its own indicatrix. A circle of three radii on the ground at 30°, 40°, projected exactly — the solid curve — against the ellipse the indicatrix predicts for it, dashed. The filled dot is the image of the circle's centre and the hollow one is the centre of area of what was actually drawn, which is not the same point. The departure runs from 3.14 per cent at 4° to 15.80 per cent at 16°, so it grows in proportion to the radius rather than to its square: halving the circle halves the relative error and does not quarter it.

The indicatrix is a limit

Tissot's ellipse describes an infinitesimal circle, and every published one is drawn finite. The error of the description falls as the radius rather than as its square — halving the circle halves the lie — and on the Robinson projection at a table entry it does not fall at all: sixty metres and six kilometres are both 1.17 per cent wrong.

distortion · Tissot The order the corrections go in, and what it costs. Setting out a 51.8 km line on the UTM zone 31N means turning a grid bearing into an azimuth to observe, and there are two corrections: the meridian convergence, 0.8669° here, and the arc-to-chord correction, 6.242″. Each bar carries the lateral offset it produces at the far end of the line, because a bearing error is a number nobody can picture and a sideways miss is the thing that misses. The exact arc-to-chord and the classical formula every manual gives differ by 0.0044″, which is 1.10 mm at the far end — small, real, and the reason the corrections have an order rather than a sum.

Setting out runs the chain backwards

Turning a design coordinate into something to observe on the ground means undoing the reduction chain, and undoing a chain reverses the order as well as the operations. Two of the corrections do not commute, and getting them the wrong way round misses by a millimetre at fifty kilometres.

practice · Reduction The minimum-distortion conformal map of an elongated region, 30° by 10°. The conformal projection of an elongated region, 30° by 10° whose scale is constant on the boundary, which is Chebyshev's criterion, obtained by fitting eight terms of a series rather than by choosing a named projection. The scale factor runs from 0.98640 to 1.00000, a spread of 1.01379; on the boundary itself the largest departure from constancy is 2.13e-7 in the log, which is what the fit achieved and not what it was told. Each dot is an interior sample shaded by its own departure from the boundary's scale.

Solving for the map instead of choosing it

Chebyshev's criterion has sat on this site since its second phase with one case it could be applied to: the spherical cap, whose answer is the stereographic projection. For any other region the site stated the criterion and stopped. It is a linear least-squares fit, and the fitted map beats every named projection over the region it was fitted to.

choosing · Condition Clairaut's theorem, and the term it drops. Clairaut's theorem says the flattening of a rotating body plus the flattening of the gravity on it equals five halves of the ratio of centrifugal to gravitational acceleration at the equator. Measured on WGS84: f = 3.3528e-3, f* = 5.3024e-3, and their sum is 8.65525e-3 against the theorem's 8.62447e-3. The residual is 3.08e-5, which is 2.74 times f² — second order, which is what a first-order theorem is entitled to be wrong by.

The flattening is not a free parameter

An ellipsoid is usually presented as two numbers somebody fitted. One of them is not free — Clairaut's theorem relates the shape of a rotating body to the gravity on it, and the relation holds on WGS84 with a residual of 3.1×10⁻⁵ — which is 2.74 times f², exactly what a first-order theorem is entitled to.

datums · Ellipsoid The conformal map onto a square face of a cube. The spherical face of a cube carried onto its flat face by a map that is conformal everywhere — the measured angular deformation over the drawn interior is 1.63e-6°, which is the arithmetic's own floor. The rings and spokes are circles and radii on the sphere, and they cross at right angles here because that is what conformal means. The map was solved for as 16 terms of a series rather than written down: the face's edge comes out straight to 0.33 per cent of its own half-width, and that residual — not the conformality — is what more terms buy. At each corner the map behaves like ζ^0.75, so the scale factor there is infinite.

A conformal map onto a face

The polyhedral ladder ended owing a conformal face map, on the grounds that it needs elliptic functions. It does not: a conformal map of the sphere is an analytic function of one conformal coordinate, so the map is a power series, choosing it is a least-squares fit — and its scale factor is infinite at the corners, which is the angle deficit arriving as a singularity.

families · Polyhedral What a solved map is, as a list of numbers. A map with no formula is a list of coefficients, and this is the list. The Chebyshev map of an elongated region, 30° by 10° has its coefficients falling by a factor of 1.5e+13 from the first to the fourteenth, so a table of a dozen numbers carries the whole projection; the conformal cube face's coefficients, marked separately, fall far more slowly because the map has a singularity at each corner. How fast this line falls is exactly how portable the map is — and neither map has a name, an inverse in closed form, or a formula anybody could quote.

A map with no formula

The solved projection has no name, no formula and no closed-form inverse. It is fourteen numbers — and the rate at which those numbers fall away decides whether a map can be shipped at all: geometrically for a smooth region, and like a power for one with corners.

choosing · Condition The error of a spherical formula is twice the flattening. The worst angular deformation of the spherical Mercator formulae applied to each body's own latitudes, against that body's flattening, on logarithmic axes. The points are measured by differencing the projection; the line is 2f radians, which is not fitted. The two agree to a third of a per cent for Mercury, the Earth and Mars, and depart by 3.3 and 5.1 per cent for Jupiter and Saturn, whose flattenings are too large for the first term to be the whole story. The site's headline number — 0.3848° on Earth — is an instance of this law rather than a fact about Web Mercator.

The same projection on a different body

A projection's distortion does not depend on the body's size at all — only on its flattening — so Web Mercator's 0.3848° of angular deformation is not a fact about Web Mercator. It is 2f radians, and the same mistake made on Mars measures 0.6765°, on Jupiter 7.68°, and on a body scaled ten times larger than Mars the identical number to twelve figures.

datums · Bodies The corner against where along the seam the feature crosses. A feature crossing the seam of a cube at right angles, drawn on each face's own map and unfolded, with the crossing moved along the edge. Every curve starts at zero, because the face's mirror through the edge's own midpoint reverses the along-edge direction and forces the shear term in the Jacobian to be odd. Away from it the gnomonic — the map with no corner at all in the rung below — reaches 44.4°, three times the equal-area map's and far past the conformal one's. The edge's half-length is 35.3°, so the right-hand end is still well inside it.

The corner is not at the midpoint

The rung below measured the corner a feature gets crossing a polyhedral seam, and found none at all under the gnomonic face map. It crossed at the edge's own midpoint every time — the one point on the edge where a face's own mirror symmetry forces the corner to vanish. Two fifths of the way to the vertex the gnomonic gives 20.15°, the equal-area map 7.28° and the conformal map under a degree, which reverses the ordering entirely.

families · Polyhedral One of these is a corner. The corner reported at the exact conformal seam, and the corner reported at a cube's gnomonic seam fifteen degrees along its edge, each measured over a shrinking arc. The gnomonic's is 21.4572° at every span from twenty-four degrees down to three — the same number to four decimals — because it is a corner. The conformal one halves whenever the span does, fitted exponent 0.990, because a tangent read from a finite chord of a curved image departs from the true tangent in proportion to the chord. It is not a corner; it is the instrument.

The exact map says the seam is smooth

The previous rung could only bound the conformal seam's corner at about two degrees, because the series it was fitted with holds its boundary condition to three parts in a thousand. An exact map exists — the stereographic projection composed with ∫dt/√(1 − t⁴) — and it settles it: the corner falls in exact proportion to the arc the tangent is read over, fitted exponent 0.99, while the gnomonic's 21.4572° is the same to four decimals at every span.

families · Polyhedral Two expansions of one integral. The worst error along the whole meridian, against the number of sine terms kept, for the series in the third flattening and the classical series in e². Both are checked against a Simpson's rule on the defining integral, which shares no algebra with either. At one and two terms they are the same number to four digits and the e² series is fractionally ahead; from the third term the n series pulls away, and at four it is 1175 times more accurate — 7.6e-8 metres against 9.0e-5.

Which small quantity the series is in

Every ellipsoidal formula in this collection is a truncated series in the third flattening, inherited from Krüger in 1912 and justified nowhere. Measured against the classical expansion in e², at four sine terms it is 1,175 times more accurate — and at one and two terms it is fractionally worse, which is not what the folklore implies.

wrong · Ellipsoid Where the condition holds, and where it was asked to. The boundary scale of a fit collocated at 20 points, drawn all the way round the boundary. The marked points are the ones the condition was imposed at, and the curve passes very near zero at every one of them; between them it does not. The largest departure on the samples is 3.92e-5 and the largest anywhere is 3.27e-4, and the second is the one the map has.

A condition imposed at points is not a condition

Nine rungs state a condition and solve it, and every solve imposes the condition at a finite set of samples because that is what a linear system is. With barely more equations than unknowns the residual the solver reports is 8.3 times too good — and refining the collocation twentyfold does not improve the map at all, it only makes the report honest.

choosing · Condition Three face maps on a cube, over a shrinking span. The corner a feature gets crossing the seam of a cube, measured by reading the tangent over an arc and then shrinking the arc by a factor of sixteen. A chord differs from a tangent in proportion to the arc, so a perfectly smooth join reports a corner that HALVES when the span halves — a slope of one on these axes. None of the three lines has a slope of one. The fitted slopes are 0.000, 0.000, -0.018, which is a flat line in each case, and a flat line is a real corner. The three differ in size and not in kind: 20.1513°, 7.2772°, 0.7687°.

The span ladder, run on all five

A recorded shortfall said the exact map's span ladder was one loop away from settling the seam on the five Platonic solids. The loop was run and the answer is the other one: the series conformal face map has a corner that does not shrink with the measurement span on any of them — 4.10° on the tetrahedron, 0.77 on the cube, 0.004 on the icosahedron — and it belongs to the truncation rather than to conformality.

families · Polyhedral What a geoid model leaves out, against the degree it stops at. The RMS of everything above the model's highest degree, from Kaula's rule — the statement that the normalised coefficients at degree n are about 10⁻⁵/n². The line is R × 10⁻⁵ ÷ n, so a model to degree 360 omits 17.7 centimetres and one to 2190 omits 2.9. Every orthometric height derived from such a model carries that as an error, and it is not quoted with the height.

The geoid model stops at a degree

Eleven essays treat the geoid as a surface that exists. Every geoid anybody uses is a series truncated at a degree, so every orthometric height derived from one carries an omission error nobody quotes with the height — eighteen centimetres at degree 360 — and the same truncation removes two thirds of the slope, which does not converge at all.

datums · Height One of these settles. The largest departure of the fitted map's boundary scale from constant — the quantity the previous rung showed the solver cannot see — against the number of collocation nodes, for the two placements. The clustered fit reaches 7.030e-5 at forty-eight nodes and returns exactly that at every count above it. The evenly spaced fit does not settle at all: it wanders by a factor of 1.43 across the same range, going up as often as down. Refining an evenly collocated fit is not convergence, and the previous rung's finding that more samples improve the report and not the map is this seen from one side.

The nodes were evenly spaced

The previous rung showed that refining an evenly collocated fit improves the solver's report and leaves the map alone. Moving the same number of nodes to the Chebyshev positions — crowded toward the corners, where a conformal map of a polygon is singular — makes the fit settle: 7.030 × 10⁻⁵ at forty-eight nodes and exactly that at every count above it, against an even fit that wanders by 43 per cent and never converges at all.

choosing · Condition

Named alongside it

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

ConformalityVerificationToleranceConvergence rateLeast-squaresClosed formEllipsoidConvergenceFlatteningPolyhedral projectionInverse problemJacobian

All concepts