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

The free function, drawn

Drawn here at the parameters it defaults to, with every essay that calls it.
The free function, drawn. The quantity the family leaves free, shown as what it decides: how long each parallel is drawn relative to the equator's. The equal-area condition C·Y′ = cos φ fixes one of C and Y from the other, so this curve IS the map. The sinusoidal's is cos φ exactly, which is why its parallels are true to scale; Mollweide's falls faster and reaches zero at the pole; Eckert IV's stops short of zero, which is its pole line. The searched member's is a curve none of the three names, and it is a curve rather than a choice among curves — which is the whole of what this rung is about.

The quantity the family leaves free, shown as what it decides: how long each parallel is drawn relative to the equator's. The equal-area condition C·Y′ = cos φ fixes one of C and Y from the other, so this curve IS the map. The sinusoidal's is cos φ exactly, which is why its parallels are true to scale; Mollweide's falls faster and reaches zero at the pole; Eckert IV's stops short of zero, which is its pole line. The searched member's is a curve none of the three names, and it is a curve rather than a choice among curves — which is the whole of what this rung is about.

It is drawn by projection-figure with show: "pseudo-shape" — 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.

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.

Three named members of the family, and one the family has no name for. The equal-area pseudocylindricals are not a list of maps. They are the solutions of one equation — C(φ)·Y′(φ) = cos φ — in two unknown functions, so one function is free and the named members are points in a space of them. Writing Y as six numbers and searching over them finds a map at 25.03° of mean angular deformation against Eckert IV's 28.08, Mollweide's 31.83 and the sinusoidal's 38.86 — 12.2 per cent better than the best map the family has a name for, and every one of the four is equal-area to the same precision. The families

A family is a function, not a list

The equal-area pseudocylindricals are the solutions of one equation in two unknown functions, so one function is free and the named members are points in a space of them. Writing that function as six numbers and searching over them finds a map at 25.03° of mean angular deformation against Eckert IV's 28.08 — and two numbers are already enough to beat every map the family has a name for.

The whole library