Concept

Projection — where it appears

A rule taking positions on a body to positions on a plane, which is what every map is drawn through. No such rule preserves every length, which is a theorem rather than an engineering limitation, and the whole subject is what each one gives up instead.

Named by 5 essays across 5 fields — each of them below, with the objects they name alongside it.

Two explanations for one residual, against the size of the region. A map in the right projection whose control coordinates are on the wrong datum, and a map fitted with the wrong projection and the right datum, both measured as a residual after the reproduction's scale, rotation and offset have been removed. The datum shift's residual is flat: it is the same 1.3e-6 at every size, because a similarity fit is very nearly what a datum shift is and it absorbs the rest. The wrong projection's grows by three orders of magnitude with the region. They are the same number below about a degree, which is where a residual stops saying anything about either of them.

A residual has more than one explanation

The method names a projection by fitting every candidate to a set of control points and taking the smallest residual. It has never been asked what else a small residual could be. A map drawn in the right projection from coordinates on the wrong datum leaves a residual of one part in a million — indistinguishable from noise, at every region size, because a similarity fit absorbs a datum shift almost exactly.

wrong · Identify

Two routes to one grid distance, 76.7 kilometres apart. The first four bars are the tape's chain: the chord between two marks, the height difference taken off it, the reduction to the ellipsoid, and the projection's own scale applied. The last is the satellite route — project both marks and subtract — which needs none of those steps because the observation already contains the positions the chain exists to supply. They agree to 0.17 millimetres, and that agreement is what makes the chain checkable at all: two routes to one number, sharing no arithmetic.

The chain the satellite does not have

The four-step reduction from a tape reading to a grid coordinate exists because a tape does not know where it is. A satellite observation does, so the same distance can be got by projecting both ends and subtracting — and the two answers agreeing to a fifth of a millimetre is the first check the chain has ever had.

practice · Reduction

One construction, one number. A ray from a point on the equatorial plane, through the sphere, onto a tangent cylinder. Where the centre sits is the whole of the family: on the axis it gives tan φ, at the far side of the sphere 2 tan(φ/2), and at infinity sin φ. The same expression with angular distance in place of latitude, and a tangent plane in place of a cylinder, gives the gnomonic, the stereographic and the orthographic. The surface decides whether the answer is read as a height or as a radius and nothing else.

The developable surface was never necessary

Eight essays on this ladder start from cylinders, cones and planes, and none of them has asked which named projection is actually a projection onto one. Fitting a one-parameter perspective construction to eleven of them: six come back exact, and the three landmarks are the gnomonic at h = 0, the stereographic at h = 1 and the orthographic at h = ∞.

families · Families

The indicatrix is a limit, and here it is being taken. The axis ratio of the image of a circle of radius ρ, measured from the picture rather than computed from the derivatives, against the ratio the derivatives give. It converges, and it converges as ρ² — each halving of the radius quarters the departure. At the smallest radius drawn the two agree to 7.6e-4 relative. This is what "the indicatrix is a limit" means when it is working: there is a number for the ratio to settle on, and it settles.

The indicatrix at a point that has none

Ten essays on this ladder take the derivative for granted. Rung three proved the indicatrix is a limit and every figure since has assumed the limit exists. There are points on the maps this collection draws where it does not, and they are not exotic: the horizon of a gnomonic map, and every corner of every polyhedral net.

distortion · Tissot

The same tolerance, applied in two orders, at 65°. The faint line is the region's boundary as built, 1025 vertices across 400 km of ground. Both pipelines were given the same tolerance of 2000 m on the ground. Simplifying in degrees and then projecting keeps 311 of them; projecting into Mercator and then simplifying keeps 129; doing it on the ground itself, which no pipeline does, keeps 129. The two drawn lines separate by 1883 m, which is 94 per cent of the tolerance that was supposed to bound the whole operation.

Simplification does not commute with the projection

A pipeline either simplifies the geometry and then projects it, or projects it and then simplifies. Both orders are in use, neither is recorded, and given the same tolerance in ground metres they keep different vertices — 129 of them on the ground, 367 in degree space at 80°, and 459 on an equal-area page.

applied · Generalise

Named alongside it

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

Conformality Gnomonic Indicatrix Scale factor Tolerance Angle deficit Anisotropy Azimuthal Baseline Chord Classification Closed form

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