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

Sinusoidal, cut into 6 lobes

Drawn here at the parameters it defaults to, with every essay that calls it.
Sinusoidal, cut into 6 lobes. six lobes, cut through the oceans so each continent stays whole. Each lobe is the same projection about its own central meridian, so each point is near a line where the shape distortion vanishes: the mean angular deformation over the mapped world falls from 38.6° uninterrupted to 17.8° here. What is given up is that the map is no longer one surface — the graticule stops at every cut because there is nothing on the other side of it.

six lobes, cut through the oceans so each continent stays whole. Each lobe is the same projection about its own central meridian, so each point is near a line where the shape distortion vanishes: the mean angular deformation over the mapped world falls from 38.6° uninterrupted to 17.8° here. What is given up is that the map is no longer one surface — the graticule stops at every cut because there is nothing on the other side of it.

It is drawn by projection-figure with show: "interrupted-map" — one member of a family of 16 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.

4 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.

Sinusoidal, cut into 6 lobes. six lobes, cut through the oceans so each continent stays whole. Each lobe is the same projection about its own central meridian, so each point is near a line where the shape distortion vanishes: the mean angular deformation over the mapped world falls from 38.6° uninterrupted to 17.8° here. What is given up is that the map is no longer one surface — the graticule stops at every cut because there is nothing on the other side of it. What each projection optimises

Giving up continuity

Cutting a map into lobes really does reduce the distortion, by a factor that can be measured. What is paid is that the map stops being one surface — and the size of the tear is the number that pictures of interrupted maps never carry.

 American polyconic. The graticule of the American polyconic projection at 30° of longitude and 15° of latitude. a different tangent cone for every parallel — true to scale along all of them, and along the central meridian. It is neither conformal nor equal-area. The families

The projections that gave up being one thing

The polyconic is built from a different cone for every parallel, which means it is built from no cone at all. It preserves nothing the usual tests look for, it has an exact property neither of them measures, and its sheets do not fit together — a defect discovered in the field rather than at the drawing board.

 What a pole line buys, and what it costs. four pseudocylindrical projections placed by the two numbers the decision moves. Across the bottom, the angular deformation averaged over the whole sphere, where the pole-line projections — Eckert IV and Robinson — are the better maps. Up the side, on a log axis, the factor by which the last parallel is stretched, where they are worse by more than tenfold: 49× at best against 4.6× at worst for a pole drawn as a point. A pole line represents one point of the globe by a line of the map, and that is the price. The families

Where a pseudocylindrical puts its error

Every projection in this family has to decide what to do at the pole, which on the globe is a point where every meridian meets. Drawing it as a point and drawing it as a line are the two answers, and the trade between them is measurable in both directions.

 A 20° shape across the antimeridian, in the space where the numbers live. Longitude runs across the page from −180° to 180°, which is where the failure is: the shape is one rectangle on the ground and two pieces in the numbering, and every operation that treats longitude as a real number sees the two. The bounding box comes out 359° wide instead of 20°, the planar area comes out 17 times too large because the shoelace encloses the complement, and the midpoint of a segment from one edge to the other lands 20015 kilometres away — the antipode of where it belongs. The true area, from the closed form, is 4,920,667 square kilometres. What a machine does with it

The antimeridian is a cut in the numbers

A twenty-degree box across 180° has a bounding box of 359.4°, a planar area seventeen times too large, and a midpoint 20,015 kilometres from where it belongs — which is the antipode, exactly. Moving the cut moves the failure and never removes it, because a circle cannot be numbered by an interval.

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