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

Where the coordinate turns back

Drawn here at the parameters it defaults to, with every essay that calls it.
Where the coordinate turns back. The vertical map coordinate against latitude, along six meridians of Craig's retroazimuthal projection. Every curve turns over inside the mapped band, so every value on the inner side of the turn is taken twice. On the marked meridian, -77.8° and -48.1° of latitude are drawn at the same height, to 2.2e-8 in map units. A projection is usually a function with an inverse; this one is a function without one.

The vertical map coordinate against latitude, along six meridians of Craig's retroazimuthal projection. Every curve turns over inside the mapped band, so every value on the inner side of the turn is taken twice. On the marked meridian, -77.8° and -48.1° of latitude are drawn at the same height, to 2.2e-8 in map units. A projection is usually a function with an inverse; this one is a function without one.

It is one of 4 figures drawn by the same construction, and the drawing above is this one with nothing changed.

1 essay calls 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.

A map for pointing, not for locating. Craig's retroazimuthal projection, centred on Mecca. The straight line from any point to the centre makes an angle with the map's vertical equal to the true initial bearing from that place to the centre — checked here at 169 points, worst disagreement 5.7e-14 degrees. The bearings printed beside each line are computed on the sphere with no projection in them. What each projection optimises

A map that cannot be read backwards

Craig's projection answers one question exactly — lay a straight edge from any place to the centre and read the compass course, right to 6 × 10⁻¹⁴ of a degree. It pays by folding: 78°S and 48°S on the same meridian are drawn at the same point, so no inverse exists and nothing else can be read off it at all.

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