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Mapping the Earth with a stacked-system of Cylindrical Equal-Area maps for better shape & scale

 
 
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Reply Mon 30 Aug, 2021 10:34 pm

I propose an innovative, original, stacked-system of Cylindrical Equal-Area map-sections. optimally sized & matched to show a hugely greater percentage of the Earth’s surface with good shape & scale, compared to all other equal-area maps.
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I call it “CEA-Stack”.
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Summary of CEA-Stack world-map proposal:
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What if there were an equal-area world map that could portray everywhere from the tip of South-America up to Glasgow & Copenhagen (5/6 of the Earth’s surface) with good shape like the Behrmann Projection’s portrayal of Africa?
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(google: “Behrmann Projection”)
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…and show everywhere between Antarctica & the middle of Iceland (90% of the Earth’s surface) with scale at least equal, in every direction, to the scale along the equator?
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Well, there is one.
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I call it CEA-Stack. It consists of several Cylindrical-Equal-Area (CEA) maps in a stack…with the Behrmann Projection mapping the entire Earth, at the middle of the stack, and with, stacked above it & below it, a more vertically-expanded CEA map duplicate-mapping only the high-lat places that Behrman doesn’t well-portray the shape of.
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That’s the CEA-Stack version that uses only one high-lat map, stacked above & below Behrmann.
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Optionally:
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If a 2nd, even more north-south-expanded, CEA map is stacked above the 1st one, duplicate-mapping only the high latitudes that the 1st one doesn’t show as well as Behrmann’s Africa, then the overall combination will show everywhere between Antarctica & the middle of Iceland (90% of the Earth’s surface) with good shape like Behrmann’s Africa.
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…and will show everywhere from Antarctica nearly up to Svalbard (97% of the Earth’s surface) with scale everywhere, in every direction, at least equal to the scale along the equator.
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Below is a more detailed description of CEA-Stack:
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Proposal of a Stacked combination of Cylindrical-Equal-Area Maps with Good Scale & Shape Over Much More of the Earth’s Surface:
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Because the low-lat portion of this proposed stack of CEA (Cylindrical-Equal-Area) maps is the Behrmann CEA projection (…because Behrmann’s Africa has good shape, unlike many equal-area maps), some comments about Behrmann’s properties are called for.
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First a few definitions used here:
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point-min-scale at a point is the smallest scale at that point.
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point-min/max-scale at a point is the ratio of largest to smallest scale at that point.
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“good scale” means point-min-scale at least equal to the map’s average scale along the equator, with the map in equatorial-aspect.
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(or, more generally, not less than the average scale across the map’s largest width (X-dimension).
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“good shape” means point-min/max scale of at least ¾. (That’s its value at Behrmann’s equator).
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(On CEA maps, the lat-band with good scale is also the lat-band in which the ratio of NS/EW scales is at least as high as would be perceived, due to foreshortening, at the nearest meridian of a globe viewed from over its equator, at distance.)
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Properties of Behrmann CEA:
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Standard parallel (at which NS & EW scales are equal, and shapes are perfect):
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Lat 30.
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(That’s near much of the Mediterranean coast of Africa, & along the Gulf Coast of the U.S.; and just below the Florida-Georgia border.)
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Half of the Earth is poleward of lat 30, and NS-compressed in Behrmann.
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Half of the Earth is equatorward of lat 30, and is EW-compressed in Behrmann.
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On Behrmann CEA, good scale and good shape both extend from lat 41.41 south, to lat 41.41 north.
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That’ about 2/3 of the Earth’s surface.
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That lat-band’s northern extent is to near Barcelona,Spain; Southern New York; Omaha, Nebraska; and Mount Shasta in California.
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Its southern extent covers all of Africa, all of Australia except for Tasmania (over which it extends partway), New Zealand’s North Island; and, in South-America, most of the way down Argentina & Chile.
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Strongly NS-expanded CEA-versions, such as Gall-Peters, Balthasart, & Square Tobler CEA, improve shape & scale at high-lat, but grossly distort low-lat shapes, resulting in very noticeable unrealism. …while also wasting vertical space with their large disproportionate exaggeration of NS scale at low-lat.
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So then, for optimal shape everywhere, why not just use a strongly NS-Expanded CEA version only where it’s desirable?:
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…at high-lat.
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…starting at lat 41.41, where Behrmann starts having point-min/max-scale less than ¾.
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Though CEA-Stack’s Behrmann map maps the entire Earth, this more NS-expanded map, stacked directly over & under the Behrmann map, would duplicately-map only the region from lat 41.41 to the pole.
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The NS-expansion of that high-lat map (with respect to Lambert CEA) would be chosen to achieve a NS/EW scale-ratio of 4/3 at lat 41/41. That’s achieved if the high-lat map multiplies Lambert CEA’s NS dimension by a factor of 2.370370370…
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That resulting high-lat map is very similar to Balthasart CEA. Its standard parallel is at latitude 49.49. …just north of Paris, and the U.S.-Canada border.
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With that stacked combination, the Earth from the tip of South-America, up to the approximate latitude of Glasgow & Copenhagen, is portrayed with point-min/max-scale of at least ¾. …i.e. with good shape (as defined above).
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That’s about 5/6 of the Earth’s surface.
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And the Earth from Antarctica to the middle of Iceland is portrayed with good scale (as defined above).
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That’s about 90% of the Earth’s surface.
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As impressive as Behrmann’s shape & scale properties are, this stacked combination, which I call “CEA-Stack” does even better.
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No equal-area single-map can achieve good shape & scale over anywhere near as large a percentage of the Earth as CEA-Stack does.
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It’s the first new use of CEA since CEA was introduced in 1772. …and arguably the first big properties-improvement in CEA since the Behrmann CEA in 1910.
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Optionally:
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Obviously, the procedure that stacks a separate high-lat map to expand the regions of good shape & scale, could be applied a 2nd time, adding, above the 1st high-lat map-section, a 2nd, upper, even more north-south-expanded, CEA map-section.
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…and thereby achieving good shape from Antarctica up to mid-Iceland (That’s 90% of the Earth).
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…and good scale nearly up to the arctic archipelago of Svalbard (That’s 97% of the Earth).
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As before, from the point where the 1st high-lat map’s point-min/max-scale drops to ¾, the land poleward from there is duplicately-mapped by the 2nd high-lat map, which (as before) is the CEA-version with NS/EW scale-ratio = 4/3 at that latitude.
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Typically a wall-map’s vertical space-availability is fairly unlimited, allowing space for all of CEA-Stack’s map-sections to map all the way to the pole.
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(But if there were a shortage of vertical space, then, when two high-lat maps are used, the 1st high-lat section could, optionally, map only to the point where the 2nd high-lat section starts. …because the whole-Earth Behrmann section is available for area-comparisons everywhere.)
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(But there’d be no reason to have that 2nd high-lat section in the Antarctic, because it would just extend the good shape & scale farther into Antarctica. …unless Antarctica is of particular interest.)
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That 2nd high-lat CEA map would multiply the Y-dimension of Lambert CEA by a factor of 4.214
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It would have a standard-parallel at lat 60.8437
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That’s north of Oslo, Stockholm & Helsinki.
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It would have, over its range of mapped latitudes, NS scale about 34% larger than that of Square Tobler CEA.
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One thing evident from images of Balthasart & Tobler CEA is that CEA-Stack shows Greenland much better than most other CEA maps do.
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If some don’t want the high-lat section(s) to have greater magnification than the Behrmann section (the greater magnification is needed to expand the region of good scale), then the high-lat sections could be shrunk in both dimensions, to achieve magnification equal to that of the Behrmann section. …giving up the scale-advantage, but keeping the shape-advantage. …resulting in a terraced map-outline.
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Formulas for Each of CEA-Stack’s Sections, for Each of Four CEA-Stack Versions:
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For each section of CEA-Stack, the formulas are, of course, the formulas for an ordinary CEA map, differing, between sections, only in their Y-Multiplication-factor, F. …and a positioning constant to place their respective sections at the correct Y-positions on the mapsheet.
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The formulas for each map-section automatically place it at the right position on the mapsheet. For a particular mapsheet using a particular CEA-Stack version, the formulas for each of the component map-sections are all to be applied on the same mapsheet. Each section’s Y-positioning on the mapsheet is automatic, via its formula’s Y-positioning constant. For each version of CEA-Stack, the formulas for its map-sections are listed together.
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The formula for each CEA-Stack map-section is just the ordinary general CEA formula (discussed immediately below), with a different value for F (the Y-multiplication factor), and with a different Y-positioning constant.
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Formulas for four CEA-Stack versions are given farther below. Listed immediately below are the four versions for which formulas are given farther below:
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1. Version 1, using one application of the CEA-Stack procedure, resulting in the use of one Y-expanded CEA projection for duplicate-mapping at high-lat.
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2. Version 2, using two applications of the CEA-Stack procedure, resulting in the use of two Y-expanded CEA projections for duplicate-mapping at high-lat.
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3. Version 3. Like version 1, but with the high-lat section shrunk to give it the same magnification as the Behrmann map-section. The scale-advantage is lost, but the shape-advantage remains.
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4. Version 4. Like version 2, but with the high-lat sections shrunk to give them the same magnification as the Behrmann map-section. The scale advantage is lost, but the shape-advantage remains.
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If the formulas below, at first-glance look time-consuming to verify the validity of, then I emphasize that the X-formula, and the 1st term of the Y-formula, are just the ordinary, familiar, unmistakable formulas for Cylindrical Equal-Area.
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The 2nd term of the Y-formula (where present) doesn’t contain either of the independent-variables Lat or Lon. It’s just a constant for Y-positioning of a high-lat map-section, to place it at the proper position in the stack.
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That 2nd term of the Y-formula, where present, is the only part of the formulas that looks time-consuming to verify. But when the formulas are entered into a program that prints or displays a map when its formulas are entered, it will be obvious that the map-sections are indeed positioned correctly--in a stack, with the sections separated by the desired separation-distance (represented as “d” in the formulas).
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(A separation of about a centimeter would probably be desirable, between the stacked map-sections.)
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General CEA formulas:
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F is the factor by which Lambert CEA’s vertical dimension is multiplied. It’s value is given for each CEA-version used in CEA-Stack.
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R is the radius of the generating globe.
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Pi is the ratio of a circle’s circumference to its radius, about 3.14159…
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Latitude is positive north of the equator and negative south of the equator.
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Longitude is positive east of the central-meridian and negative west of the central meridian.
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Y = F*R*sin(lat)
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X = (Pi*R) * Lon/180
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(…because the distance from mid-map to edge is half of the map’s width, and maps half of the Earth’s circumference. …half of 2*Pi*R)
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But it seems to me that, the map’s intended width, W, is a more convenient distance-referent than R.
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W = 2*Pi*R
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R = W/(2*Pi)
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Substituting W/(2*Pi) for R:
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Y = (W/(2*Pi)) * F * sin(Lat)
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X = (W/2)*Lon/180 = W*Lon/360
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Formulas for the 4 particular versions:
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Version 1:
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Behrmann section:
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F1 = 4/3
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For Lat = -90 to +90, and for Lon from -180 to +180:
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Y = F * (W/(2*Pi)) * sin(Lat)
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X = W * Lon/360
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Northern hi-lat section:
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F2= 2.370370370…
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Lat1 = 41,40962211
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d = desired separation-distance between the vertically-stacked map-sections
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For Lat = Lat 1 to 90 and for Lon from -180 to +180:
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Y = F2*(W/(2*Pi)) * (sin(Lat) – sin(Lat1)) + F1*(W/(2*Pi)) + d
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X = W*Lon/360
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1st Southern high-lat section:
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F2 = 2.370370370…
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Lat1 = -41.40962211
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For lat = Lat1 to -90, and Lon from -180 to +180.
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Y = F2*(W/(2*Pi)) * (Sin(Lat) – Sin(Lat1)) – F1*(W/(2*Pi)) - d
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X = W*Lon/360
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Version 2:
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(Identical to Version 1 except for the 2nd high-lat section)
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Behrmann section:
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F1 = 4/3
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For Lat = -90 to +90, and for Lon = -180 to +180:
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Y = F1 * (W/(2*Pi)) * sin(Lat)
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X = W * Lon/360
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1st Northern hi-lat section:
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F2 = 2.370370370…
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Lat1 = 41,40962211
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d = desired separation-distance between map-sections
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For Lat from Lat1 to 90, and for Lon from -180 to + 180:
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Y = F2*(W/(2*Pi)) * (sin(Lat) – Sin(Lat1)) + F1*(W/(2*Pi)) + d
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X = W*Lon/360
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1st Southern high-lat section:
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(As mentioned, this section shows down to the tip of South-America with good shape & scale. But, unless it’s desired to examine routes or position-relations near Cape-Horn or beyond, this section (and of course the 2nd southern high-lat section too) might not be needed. As already mentioned, Behrmann shows South-America with good shape & scale most of the way down Argentina & Chile. …and southern Tasmania & New Zealand’s South-Island are very near to Behrmann’s zone of good shape & scale.)
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F2 = 2.370370370…
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Lat1 = -41.40962211
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For Lat from Lat 1 to -90, and for Lon from -180 to +180:
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Y = F2*(W/(2*Pi)) * (Sin(Lat) – Sin(Lat1)) – F1*(W/(2*Pi)) – d
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2nd Northern hi-lat section:
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F3 = 4.2139917689
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Lat1 = 41.40962211
Lat2 = 55.771133669
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From Lat from Lat2 to +90, and for Lon from -180 to +180:
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Y = F3*(W/(2*Pi)) * (Sin(Lat) – Sin(Lat2)) + (W/(2*Pi)) * (F1 + F2*(1 – sin(Lat1))) + 2*d
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X = W*Lon/360
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2ndst Southern high-lat section:
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(This map-section probably isn’t needed unless precise examination of Antarctica is needed.)
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F3 = 4.2139917689
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Lat1 = -41.40962211
Lat2 = -55.771133669
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For Lat from Lat2 to -90, and for Lon from -180 to +180:
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Y = F3*(W/(2*Pi)) * (sin(Lat) - sin(Lat2)) – (W/(2*Pi)) * (F1 + F2*(1+sin(Lat1))) -2*d
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(Note: (1-sin(Lat1)) has changed to ( 1+sin(Lat1)), because Lat1 is now negative, but the term that (1 – sin(Lat)) is part of is now being subtracted, and the magnitude of what’s being subtracted should be the same as what was previously (for the Northern-Hemisphere) added.
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When Lat1 turns negative, then 1+Lat1 is the same as what 1-Lat1 was when Lat1 was positive. Doing it otherwise would have required more sign-changes in the Southern-Hemisphere.)
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For brevity here, instead of listing the formulas for versions 3 & 4, I’ll just say:
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For all map-sections, for version 3 & 4, multiply the X-formula, and the 1st term of the Y-formula, by (3/4) in the 1st high-lat map-section…and by (9/16) in the 2nd high-lat map-section.
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…and, in the 2nd high-lat map-sections, in the 2nd Y-term, multiply F2*(1+sin(Lat1)) by ¾, changing it to: F2*(1+sin(Lat1)) * (3/4). …a necessary adjustment of the 2nd section’s Y-position, due to the lower height of the 1st high-lat map-section.
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(I recommend Version 1 or 2, instead of 3 or 4, because Versions 1 & 2 improve scale, in addition to shape. But Versions 3 & 4 are offered in case some people want the magnification to be uniform across all map-sections. With them, shape is still improved, but not scale.
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I emphasize that, in versions 1 & 2, area-comparisons can be made in the Behrmann map-section, and in any other map-sections that contain both places that one wants to compare.)
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