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fmoretzsohn
(Stranger )
03/09/04 02:08 PM
 Map projections

I am working on the distribution of marine invertebrates in the Indo-Pacific, and using ArcGIS 8.3. I want to be able to measure the area of the distribution of several species to compare those with a wide distribution to those with restricted distribution. I am looking for an equal area map projection that is optimized for the tropical Indo-Pacific (if there is something like that). In ArcMap there are many different projections available, including some equal area ones, but they don't produce the exact same results. The one that I found most useful, thus far, is called World Cylindrical Equal Area (in ArcGIS).

Can anyone recommend an equal area map projection suitable for the tropical Indo-Pacific area?

Thank you.
Fabio Moretzsohn

scrabblehack
(Stranger )
05/31/06 07:49 AM
 Re: Map projections [re: fmoretzsohn]

Could you be a bit more specific by what you mean by "suitable?"

It occurs to me that the limiting cases for equal area projections are (1) Albers Equal Area (based on a cone),
(2) Cylindrical Equal Area (based on a cylinder), and (3) Lambert Azimuthal Equal Area (based on a plane).

CEA has the advantage (or disadvantage) of maintaining perpendicular lat/lon lines. LAEA has the advantage of maintaining true direction from the projection point.
Albers, using the 1/6 rule, is usually pretty close to LAEA but you can get the whole world rather than just a hemisphere.

That may not be a concern in your case.

MichaelOssipoff
(Stranger )
03/10/14 12:52 PM
 Re: Map projections [re: fmoretzsohn]

In regards to the post about an equal-area map for the Indo-Pacific region, why not use an obliqe local Aitoff-Hammer map?

Aitoff-Hammer is a projection for mapping the entire Earth, but it could also be used for any oblong-shaped smaller region, such as the Indo-Pacific region.

Aitoff-Hammer has a relatively symmetrical distortion-pattern. It's made by laterally expanding the Lambert Azimutha Equal Area projection.

An oblique Aitoff-Hammer would seem the most distortion-minimizing way to map the oblong-shaped region you spoke of.

Aitoff-Hammer is often just called "Hammer", and I'll so call it here henceforth, for brevity.

How Hammer is constructed:

Start with an oblique Lambert Azimuthal Equal Area projection. Say it's in equatorial aspect. Expand the map in the east-west dimension, by any factor, F, that you choose. In other words, multiply all east-west distances by F.

Now, re-label the meridians so that their labeling will be in keeping with the new east-west extent of the map. In other words, if F is 2, and you've doubled all of the east-west distances, then also double all of the longitude values of the meridians.

When Hammer is used as a world map, you start with a Lambert Azimuthal Equal Area map of half of the world, in equatorial aspect. The F value used is 2. That gives a 2:1 elliptical map of the world.

But:

1. The map needn't be in equatorial aspect. Of course it could be centered on any place in the world, in any orientation.

2. The map needn't be a map of the whole world. The above-described process for making Hammer from Lambert Azimuthal Equa Area could be used for any size region as well. And, of course F needn't be 2. It could be whatever value fits the shape of the region you want to map.

I haven't heard of Hammer being used for mapping a region smaller than the entire Earth, but it could be so used, as described above.

Hammer was introduced in 1892, based on an idea introduced by Aitoff 3 years previous. (Aitoff applied the east-west expansion to the Azimuthal Equidistant Projection).

Michael Ossipoff

MichaelOssipoff
(Stranger )
03/10/14 01:53 PM
 Re: Map projections [re: fmoretzsohn]

I said that I've never heard of Aitoff-Hammer being used to map a region smaller than the entire Earth. I should admit that I can't guarantee that, for a smaller region, the procedure for making Hammer from Azimuthal Equal Area remains equal-area.

If not, then the ordinary Azimuthal Equal Area (AEA), centered on the middle of your region, is probably the best suggestion that I know of for your purpose.

I say "probably" because, if you're mapping a quite skinny region, then there's a point at which an oblique cylindrical Equal Area map would give less maximum shape distortion. But, if only the ocean regions are important for your mapping purpose, then of course shapes aren't as important anyway.

Even if the AEA-to-Hammer process retains equal-area, an ordinary oblique Azimuthal Equal Area map might be easier, and almost just as good if your region isn't too oblong.

Because I can't find the button for starting a thread, is it ok if I add a comment on another topic here (but still about equal area maps)?:

Maps could be rated on how well they fill a rectangular space.  ...on how much of their "circumscribing rectangle" they fill. In other words, since the map will be on a rectangular page or map-sheet, and because wall-space for putting up a map is typically rectangularly-bounded, it's of interest how well a map fills that rectangular space in which it fits.

The more it fills that space, the more area the map can have. That means that, on the average, the map can have better resolution, show more detail, have more space for writing, etc.

I call that attribute "space-efficiency". A rectangular map has a space-efficiency of 1.0

Other space-efficiencies:

Circular or elliptical maps have a space-efficiency of pi/4 (.785)

The sinusoidal world map has a space-efficiency of 2/pi (.636)

Any rectangular map has a space efficiency of 1.0

Eckert III and Eckert IV's space-efficiency is about .89

The Quartic Authalic world map has a space-efficiency of 4/5, or .8

The Quartic Authalic is a pseudocylindrical world map, bounded by quartic (4th degree) power functions. It's constructed by starting with a parallel-spacing identical to that of the central meridian of the Azimuthal Equal Area, and extending the parallels out, at each latitude, to widths that make the map equal-area. That results in the outer boundaries of the map being 4th degree power functions, hence the map's name.

It's interesting that the Quartic Authalic's space-efficiency is almost the same as that of Hammer-Aitoff (Quartic Authalic's space-efficiency is just very slightly larger--.8 vs .785)

Anyway, a power function can be chosen that will give a space-efficiency about equal to that of the Eckert III and Eckert IV:

That would be achieved by a 7th or 8th degree power function. The map would be rather squarish, with rounded corners. It would have about the same space-efficiency as that of of Eckert III and Eckert IV.

Where that 7th or 8th degree map would differ from Eckert III and Eckert IV is that its top and bottom edges begin sloping equator-ward immediately with departure from the central meridian, even though slightly, at first--whereas, with Eckert III and Eckert IV, the top and bottom of the map are quite flat, over the centreal half of the map's width.

I mention that because I don't know if a 7th or 8th degree map would be a good substitute for Eckert IV (an equal-area world map).

Michael Ossipoff

MichaelOssipoff
(Stranger )
03/11/14 03:37 AM
 Re: Map projections [re: fmoretzsohn]

1. In my most recent post, I questioned whether "Local Hammer" retains Hammer's equal-area property. Of course it does. Local Hammer just amounts to using the central part of Hammer, for a region smaller than the entire Earth.

As I said, the only difference is that instead of always using a Hammer expansion factor (denoted by "F" in my previous posts here), you'd choose F to match the shape of the region being mapped.

2. By tropical Indo-Pacific, I assume that you mean the tropical Pacific, Indonesia region, and Indian Ocean. A few comments:

1. That region is so long and skinny that, even though Local Hammer might be better than Cylindrical Equal Area (CEA), CEA would be nearly as good, and so you might as well continue to use CEA for that mapping project.

In fact, that seems to remain true even if, by tropical Indo-Pacific, you're referring only to the tropical part of the Pacific, and the Indonesia region (without the Indian Ocean).

2. Of course the mapping of only the tropical regions makes the map long and skinny. Additionally, it greatly simplifies things, with CEA, because you'd be using CEA in its standard, equatorial aspect.

Of course the same would be true if you used Local Hammer. But CEA is simpler, and, as you've found, CEA is already available, ready-made.

3. So, bottom-line:

Because of the long skinny shape of the region that you're mapping, I recommend Cylindrical Equal Area (CEA), due to its low distortion for such a shapeed region, and due to its simplicity and ready-made availability.  ...even if, strictly-speaking, Local Hammer could somewhat reduce the maximum shape distortion (the shape distortion of a CEA map of such a long and skinny region would be slight anyway).

Michael Ossipoff

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