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

What each extra degree of freedom in the family is worth

Drawn here at the parameters it defaults to, with every essay that calls it.
What each extra degree of freedom in the family is worth. The best equal-area pseudocylindrical the family holds, against how many numbers its shape function is allowed. One number is the sinusoidal stretched — still equal-area, because the two axes are scaled by reciprocal factors, and not the same angles, because an anisotropic scaling is not a similarity — and it reaches 38.36° against the unstretched sinusoidal's 38.86. Two numbers reach 27.04, which already beats every named member. Six reach 25.03. The dashed lines are the three named maps, and the family passes the last of them between its first and second free number.

The best equal-area pseudocylindrical the family holds, against how many numbers its shape function is allowed. One number is the sinusoidal stretched — still equal-area, because the two axes are scaled by reciprocal factors, and not the same angles, because an anisotropic scaling is not a similarity — and it reaches 38.36° against the unstretched sinusoidal's 38.86. Two numbers reach 27.04, which already beats every named member. Six reach 25.03. The dashed lines are the three named maps, and the family passes the last of them between its first and second free number.

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

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