Some time ago, I asked about what i called "Local Aitoff", which would just amount to using the middle part of an Aitoff map to map an oblong region on the Earth, such as a continent. Of course, the Aitoff expansion-factor could be chosen for the shape of the continent, instead of using the usual expansion factor of 2. Of course the only part of the map that would be shown would be the middle part that contains the continent.
But now, my question is: What about a "Local Hammer"? For an equal-area map of an oblong region, like the Americas, or South America, or Africa, if you wanted equal-area, then why not make an oblique or transverse Hammer map, with the continent at the map's middle. Of course the only part shown would be the middle part that contains the continent.
As before, the expansion factor would be chosen to match the continent's shape, instead of using the usual Hammer expansion factor of 2.
If "If Local Aitoff" would reduce the scale max/min for mapping an oblong continent (as compared to Azimuthal Equidistant), then wouldn't "Local Hammer" likewise reduce the scale max/min for mapping an oblong continent (as compared to Azimutal Equal Area)?
Has that been used?
If not, is there any reason why it wouldn't be a good equal-area map for oblong continents, with lower scale max/min than that of the ordinary azimutal equal area or those continents?
On another topic, maybe it's of interest how well a world map fills a rectangular space. ...because book-pages and map-sheets are rectangular, and because wall-space for a map is typically rectangularly-bounded.
The percentage of a world map's circumscribing-rectangle that the map fills--I'll call that the map's "space-efficiency".
That quantity seems relevant, because, for a given circumscribing rectangle, the greater the map's space-efficiency, the bigger the map's area can be. That means more detail can be shown, with more resolution, and more words can be printed on the map, etc.
So, rectangular world maps have a space-efficiency of 1.0
The sinusoidal seems to have the lowest space-efficiency, at 2/pi, or .637
Between those,
Circular and elliptical maps have pi/4, or .785
The Quartic Authalic (it seems to me) has 4/5, or .8
Eckert III and Eckert IV have .893
If I was correct, above, the Quartic Authalic has slightly more space-efficiency than does Hammer-Aitoff. That seems surprising, becuase of its more sinusoidal-like shape. Did I make an error when I said that the Quartic Authalic's space-efficiency is .8?
Anyway, the question occurred to me, what pseudocylindrical map bordered by a power-function would have the same space-efficiency as Eckert IV?
It seems to me that a 7th or 8th degree power function would achieve that. ...would have about the same space-efficiency as the Eckert IV. But I don't know if it would be any better than Eckert IV in any way (or as good).
Mike
Mike
Would Hammer be good for mapping a continent?
-
RogerOwens
- Posts: 403
- Joined: Sun Feb 02, 2014 8:24 pm
Re: Would Hammer be good for mapping a continent?
These uses crop up occasionally, though you see more applications of Mollweide than Hammer or Aitoff. The customization via parameterization you mention isn’t really that effective. The region of low-distortion is diamond-shaped regardless of the parameterization; you just change its width and length. Unless the region you’re dealing with is diamond-shape, too, then these are not good projections to use. See here and here, since Hammer approaches quartic authalic as w drops from ½ to 0.RogerOwens wrote:But now, my question is: What about a "Local Hammer"? For an equal-area map of an oblong region, like the Americas, or South America, or Africa, if you wanted equal-area, then why not make an oblique or transverse Hammer map, with the continent at the map's middle. Of course the only part shown would be the middle part that contains the continent.
As before, the expansion factor would be chosen to match the continent's shape, instead of using the usual Hammer expansion factor of 2.
If "If Local Aitoff" would reduce the scale max/min for mapping an oblong continent (as compared to Azimuthal Equidistant), then wouldn't "Local Hammer" likewise reduce the scale max/min for mapping an oblong continent (as compared to Azimutal Equal Area)?
Not sure without some calculation. It’s not obvious to me which uses the rectangule more efficiently. I do think it’s useful to think in those terms, but space efficiency is just one of many factors that might affect projection choice, so a bit more or a bit less doesn’t seem like it would make a useful difference.RogerOwens wrote:On another topic, maybe it's of interest how well a world map fills a rectangular space. ...because book-pages and map-sheets are rectangular, and because wall-space for a map is typically rectangularly-bounded.
The percentage of a world map's circumscribing-rectangle that the map fills--I'll call that the map's "space-efficiency". …
If I was correct, above, the Quartic Authalic has slightly more space-efficiency than does Hammer-Aitoff. That seems surprising, becuase of its more sinusoidal-like shape. Did I make an error when I said that the Quartic Authalic's space-efficiency is .8?
Why a polynomial? (I assume that’s what you mean by a “power function”). But yes, you could approximate an Eckert IV quite closely with polynomials. The outer meridians are semicircles and the rest are semiellipses. Techniques for approximating elliptical arcs by means of polynomials are well studied.RogerOwens wrote:Anyway, the question occurred to me, what pseudocylindrical map bordered by a power-function would have the same space-efficiency as Eckert IV?
— daan
-
RogerOwens
- Posts: 403
- Joined: Sun Feb 02, 2014 8:24 pm
Re: Would Hammer be good for mapping a continent?
Yes, I can see why Local Hammer or Local Aitoff nearly always wouldn't accomplish anything. Better to just use the azimuthal.
I'll check out that link.
True, space-efficiency often or usually wouldn't be a main consideration for the choice of a map projection. And, even when it is, small differences in it could be outweighed by other considerations. I was considering the situation with very busy and cluttered data-maps, showing many kinds of data on the same map.
Why a polynomial? (I assume that’s what you mean by a “power function”).
By "power function", I just mean a 1-term function of the form Ax^p, where p can be any real number. For the power-function pseudocylindrical maps i was considering, p would be a positive number, probably greater than 1, probably at least 2 or more.
So, the parabolic pseudocylindirical, and the quartic pseudocylindrical would be examples of such a map, with p = 2, and p = 4
I think that, if p = 8.32, a power-function-bounded pseuocylidrical map would have the same space-efficiency as Eckert IV.
I'd said that I don't know if such a map would be in any way better than Eckert IV, but it would have one advantage over Eckert IV: Its poles would be points instead of lines, and that should make Antarctica look a lot more realistic.
Eckert IV has two big aesthetic disadvantages: Distortion of Africa and Antarctica. The power-function pseudocylindrical probably fix the Antarctica distortion, but not the Africa distortion.
Of course, if someone wants equal-area, and high space-efficiency or low shear-distortion, then one has to accept a skinny Africa. (or else a flattened Europe and an unrealistically wide aspect-ratio)
I read of Tobler's Hyperelliptical map, and looked at it. It shows Antarctica a lot more realistically than does Eckert IV.
It additionally has the advantage of gently-curved border at the poles, like Hammer or Mollweide, rather than the pointed poles that a power-function pseudocylindrical would always have, at least to some small degree. I don't know if the very slight pointedness of the p = 8.32 map would look bad though.
Mike
I'll check out that link.
True, space-efficiency often or usually wouldn't be a main consideration for the choice of a map projection. And, even when it is, small differences in it could be outweighed by other considerations. I was considering the situation with very busy and cluttered data-maps, showing many kinds of data on the same map.
Why a polynomial? (I assume that’s what you mean by a “power function”).
By "power function", I just mean a 1-term function of the form Ax^p, where p can be any real number. For the power-function pseudocylindrical maps i was considering, p would be a positive number, probably greater than 1, probably at least 2 or more.
So, the parabolic pseudocylindirical, and the quartic pseudocylindrical would be examples of such a map, with p = 2, and p = 4
I think that, if p = 8.32, a power-function-bounded pseuocylidrical map would have the same space-efficiency as Eckert IV.
I'd said that I don't know if such a map would be in any way better than Eckert IV, but it would have one advantage over Eckert IV: Its poles would be points instead of lines, and that should make Antarctica look a lot more realistic.
Eckert IV has two big aesthetic disadvantages: Distortion of Africa and Antarctica. The power-function pseudocylindrical probably fix the Antarctica distortion, but not the Africa distortion.
Of course, if someone wants equal-area, and high space-efficiency or low shear-distortion, then one has to accept a skinny Africa. (or else a flattened Europe and an unrealistically wide aspect-ratio)
I read of Tobler's Hyperelliptical map, and looked at it. It shows Antarctica a lot more realistically than does Eckert IV.
It additionally has the advantage of gently-curved border at the poles, like Hammer or Mollweide, rather than the pointed poles that a power-function pseudocylindrical would always have, at least to some small degree. I don't know if the very slight pointedness of the p = 8.32 map would look bad though.
Mike
-
RogerOwens
- Posts: 403
- Joined: Sun Feb 02, 2014 8:24 pm
Re: Would Hammer be good for mapping a continent?
Let me clarify what I meant by power-function pseudocylindrical:
In the funtion Ax^p,
x is the independent variable.
A and p are constants that can have any fixed real value.
...but, for most useful maps, p would probably be > 1, and probably >= 2.
The origin, for that function, is at the eastern end of the equator.
The equator is the y-axis.
The x-axis is a north-south line through the origin.
Craster's Parabolic, and Adams' Quartic are two examples of a power-function pseudocylindrical.
As I was saying, if p = 8.32, the map's space-efficieny equals that of Eckert III and Eckert IV.
When I last posted on this subject, I thought that Tobler's Hyperelliptical pseudocylindrical looked better than Eckert, but I was looking at a small image of the Tobler.
But, from a larger image of it, graticule quadrangles bordering the equator and the central meridian appeared the same shape as they have in Eckert IV. If there was any shape-difference, it was imperceptible.
But the Tobler Plainly had less space-efficiency and more shear at the outer meridians between latitudes 30 and 60.
So, to me, that means that Tobler loses to Eckert.
But Tobler's better Antarctica means that Tobler could be considred a compromise that gains that advantage at the cost of other desired properties.
A Power-Function map with p = 8.32, for .893 space-efficiency, like that of Eckert IV would still have Eckert IV's skinny Africa, but, with the point-pole, Antarctica might be portrayed better with the Power-Function.
Anyway, having a point-pole, instead of a pole-line, would, of itself, be a big improvement on Eckert.
The Power-Fucntion p=8.32 map could be made in an equal-area version and a linear (equidistant parallels) version.
Michael Ossipoff
In the funtion Ax^p,
x is the independent variable.
A and p are constants that can have any fixed real value.
...but, for most useful maps, p would probably be > 1, and probably >= 2.
The origin, for that function, is at the eastern end of the equator.
The equator is the y-axis.
The x-axis is a north-south line through the origin.
Craster's Parabolic, and Adams' Quartic are two examples of a power-function pseudocylindrical.
As I was saying, if p = 8.32, the map's space-efficieny equals that of Eckert III and Eckert IV.
When I last posted on this subject, I thought that Tobler's Hyperelliptical pseudocylindrical looked better than Eckert, but I was looking at a small image of the Tobler.
But, from a larger image of it, graticule quadrangles bordering the equator and the central meridian appeared the same shape as they have in Eckert IV. If there was any shape-difference, it was imperceptible.
But the Tobler Plainly had less space-efficiency and more shear at the outer meridians between latitudes 30 and 60.
So, to me, that means that Tobler loses to Eckert.
But Tobler's better Antarctica means that Tobler could be considred a compromise that gains that advantage at the cost of other desired properties.
A Power-Function map with p = 8.32, for .893 space-efficiency, like that of Eckert IV would still have Eckert IV's skinny Africa, but, with the point-pole, Antarctica might be portrayed better with the Power-Function.
Anyway, having a point-pole, instead of a pole-line, would, of itself, be a big improvement on Eckert.
The Power-Fucntion p=8.32 map could be made in an equal-area version and a linear (equidistant parallels) version.
Michael Ossipoff