Györffy's minimum-distortion pseudocylindrical projection

[PeteD]

In Györffy's paper Minimum-distortion pointed-polar projections for world maps by applying graticule transformation, it says "It can be proved that the minimum-distortion pseudocylindrical projections have a true-scale central meridian" and then cites Györffy's earlier paper Some Remarks on the Question of Pseudocylindrical Projections with Minimum Distortions for World Maps, which unfortunately I can't access.

I'd like to know what this minimum-distortion pseudocylindrical projection looks like and how Györffy arrived at this result. Does anyone have access to this paper, and if so, would you be happy to share it?

[Atarimaster]

I'd like to know what this minimum-distortion pseudocylindrical projection looks like and how Györffy arrived at this result. Does anyone have access to this paper, and if so, would you be happy to share it?

Regrettably, I don’t have the paper but regarding the appearance: Earlier in Minimum distortion pointed-polar projections Györffy says:
“pseudocylindrical with minimized distortion, labelled as ‘version (c)’ later, formulae in Györffy, 2016, p. 264”.
So I think that the projection he refers to in your quote is the Györffy C.

Kind regards,
Tobias

[Milo]

Minimum distortion according to which metric?

Also, "true-scale central meridian": doesn't that just mean the sinusoidal projection? Unless I'm misunderstanding what "true scale" means, that should be the only one.

[PeteD]

So I think that the projection he refers to in your quote is the Györffy C.

OK, but Györffy C is constrained to have true scale along the central meridian by setting c₅ = 1, c₆ = 0 and c₇ = 0, whereas Györffy says he's proved that minimizing distortion in pseudocylindrical projections gives true scale along the central meridian. I'd be very interested in seeing this proof.

[PeteD]

Minimum distortion according to which metric?

Presumably the Airy–Kavrayskiy criterion.

Also, "true-scale central meridian": doesn't that just mean the sinusoidal projection?

I think he means true scale along the central meridian but not necessarily perpendicular to it, so any projection where x = 0 and y = phi along the central meridian: Apian II, Eckert III and V, Winkel I and II, Wagner III and VI, Kavrayskiy VII etc.

[mapnerd2022]

On a related note, what criterion/which metric did John Snyder use when he made his two minimum-error pseundocylindricals?

[Milo]

I think he means true scale along the central meridian but not necessarily perpendicular to it,

Ah right. The sinusoidal projection would be the only equal-area one meeting that criterion (and preserving scale both along and perpendicular to the central meridian would necessarily result in an equal-area projection), but in principle anything between sinusoidal and equirectangular could work as a compromise projection.

[rschmunk]

Regrettably, I don’t have the paper but regarding the appearance: Earlier in Minimum distortion pointed-polar projections Györffy says:
“pseudocylindrical with minimized distortion, labelled as ‘version (c)’ later, formulae in Györffy, 2016, p. 264”.
So I think that the projection he refers to in your quote is the Györffy C.

Györffy (2016) present equations for six sample projections, progressing from higher values of his overall distortion parameter E_K² down to the lowest. The sixth and final of these, which appears on page 264 is indeed the Györffy (c).

[PeteD]

Györffy (2016) present equations for six sample projections, progressing from higher values of his overall distortion parameter E_K² down to the lowest. The sixth and final of these, which appears on page 264 is indeed the Györffy (c).

Thanks for the confirmation.

[PeteD]

On a related note, what criterion/which metric did John Snyder use when he made his two minimum-error pseundocylindricals?

I don't know, but from the skinny Africa, it looks more Airy than Airy-Kavrayskiy to me. Deakin definitely used the Airy criterion for his minimum-error equal-area pseudocylindrical, which looks fairly similar, though he fixed the axial ratio at 2, which means Africa doesn't get quite as skinny as in Snyder's projections.

I'd be very interested if anyone knows the answer or can provide formulae for Snyder's projections.