Tobias--
For an equal-area world map needed for precise observations and measurements about places close together, or for relatively-distant viewing, I'd choose Hatano, or maybe Urmayev II.
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Regarding Hatano:
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Yes, I like the idea of asymmetric projections, given the fact that landmasses are unequally distributed over the hemispheres and that on many maps, distortions of the landmasses only are more interesting than across the entire globe. That’s why I like the Hatano, or daan’s 2011 projection.
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Yes, and its longer pole-lines improve the min(min scale) in (say) the +/- 70 latitude band.
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That’s also why I still think that Canters W10 to W14 (Dr. Böhm’s naming) or Fig. 5.13 to Fig. 5.17 (numbers of figures in Canter’s textbook) or "Low-error polyconic projection obtained through non-constrained optimisation" to "Low-error polyconic projection with twofold symmetry, equally spaced parallels and a correct ratio of the axes" (Canter’s descriptive but somewhat awkward naming) would make a great addition to Geocart.
W13/W14 are, in my opinion, nice projections for general use world maps
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I’ll check those out. I’ve been looking at prospects for line-pole equal-area pseudocylindricals with better min(min scale) in the +/- 70 lat band. (lat 70 is close to the northern limit of West-European continental land. Of course in the south the inhabited land doesn’t get that near-polar).
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…for use when relatively distant viewing, or examination of small regions or close-together places is needed. Of course those things aren’t generally needed, outside of classrooms, which is why I best like Mollweide for an equal-area world map.
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And some improvement in min(min scale) in +/- lat 70 can be gotten by using Mollweide in two circles instead of in one ellipse. …Mollweide interrupted on 2 meridians instead of just one.
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And of course there’s another thing:
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If you’re interested in precise measurements or observations in a tiny region, then you probably don’t need for the same map to show accurate area relations over the whole map. In that case, then it’s relevant that Mercator and the popular compromise-Cylindricals have a higher min(min scale) in the +/- 70 band. …and a high mean min(min scale).
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…facilitating close measurements and observations in tiny regions or between closely-spaced places.
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So, other than for distantly-viewed classroom equal-area maps, a Mollweide and a Mercator serve the needs for equal-area, conformality (for its uniform scale in all directions at each point), and precise measurements and observations in small regions.
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Regarding Urmayev:
Ummm, probably just a typo, but Urmayev II isn’t equal-area
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I just assumed that Urmayev II was equal-area. …maybe because it’s relatively close to it. Looking at Urmayev I and II at GeoCart, I preferred the look of Urmayev II, because of its better-shaped Greenland. I just assumed that Urmayev II was equal-area.
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If Urmayev I, and not Urmayev II, is equal-area, then I’d have to re-assess my judgment about the choice of Urmayev as a high min(min scale) map in lat +/- 70 whose central-meridian shapes aren’t too bad.
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… and, by the way, I think Urmayev I actually is Wagner I, because Wagner already did provide the means to change the length of the pole line (and the ratio of the main axes on top of that), at least in his 1949 textbook (as I’ve said before, I’ve never read the original introduction of 1932).
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I didn’t know that. In fact, I didn’t have any information about the Urmayev maps, other than the images. So Urmayev I is sinusoidal at its edges?
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I guess what I was considering was the merits of an elliptical-edges flat-polar equal-area pseudocylindrical with a larger (pole-line/equator) ratio. …if there’s one that greatly improves min(min scale) in lat +/- 70, but without unacceptable central-meridian shape-proportions.
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Do you know of such a map, with (pole-line/equator) ratio higher than those of Hatano (but less than Cylindrical)?
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RogerOwens wrote:
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No doubt Huffnagel is great for PhD mathematicians. I doubt that it can be of much value or interest to the rest of us, as a published map.
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I have to disagree here.
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Especially if you’re not interested in the mathematics
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True, one can be interested instead in the
information in the maps. After all, the information is there, and of interest in itself, regardless of how the map was made.
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But, as an additional interesting subject, I just mention that all the equal-area pseudocylindricals with elliptical (including circular) edges…
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…including Mollweide, Eckert IV, Wagner IV, and Hatano…
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…have construction-derivation explanations that are explainable with reference only to geometry. …where the whole derivation is visibly shown on a geometrical diagram.
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The geometrical derivation of Cylindrical Equal Area can be part of the geometrical derivation of the elliptical-edge equal-area pseudocylindricals, making a derivation of the ellipse-edge equal-area pseudocylindricals that’s entirely geometrical.
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Or, alternatively, it can also be geometrically shown that sine(x) is the function that has cosine(x) as its instantaneous rate-of-change. …as part of an alternative geometrical demonstration for the derivation of the elliptical-edge equal-area pseudocylindricals.
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My point is just that, as an additional interesting subject, geometrical derivation of the ellipse-edge equal-area pseudocylindricals is available, accessibly-visible as geometric diagrams.
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No one should believe that the derivation of those maps is accessible only to mathematicians.
And of course it isn’t even necessary to actually
derive the formulas, though they could be derived via the geometrical diagrams mentioned above. Merely briefly cursorily looking-over the geometrical diagram derivation would be enough to satisfy oneself that one had seen how the derivation is done.
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I’m not saying that you should be interested in that, or that you need it—because of course the maps’ information is available without pursuing that derivation subject. I’m just saying that, if you’re curious about or interested in the derivation, it’s accessible.
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Of course, aside from that, it’s also true that Mercator constructed his map numerically, by spacing each next nearby parallel so as to make the north-south scale equal to the east-west scale at that latitude.
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So that construction method was good enough for that important nautical navigational map.
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Of course, his same numerical construction method can be used to construct any equal-area pseudocylindrical, spacing each next nearby parallel so as to make the north-south scale proportional to the reciprocal of the east-west scale at that latitude. (e.g. If E-W scale is double the equatorial scale, then make the N-S scale half the equatorial scale) …meaning that those maps have a construction-method that we all know about.
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Michael Ossipoff
