[Proj] Re: (Ossipoff's linear projections)

[strebe]

In other words, a "linear" projection is a pseudocylindric projection with equally spaced parallels.

There is no limit to how many such projections could be created. They follow the form:

x = λ • f (φ)
y = k • φ

where λ is longitude and φ is latitude and k is some arbitrary constant and f is some arbitrary function of (only) latitude. On the sphere, then, scale along parallels (also the horizontal scale factor) would be

f (φ) / cos (φ)

and scale along meridians would be

√(λ² • [∂f/∂φ]² + k²)

with a vertical scale factor of k.

Regards,
-- daan Strebe


On Jun 27, 2008, at 12:48:08 PM, "Michael Ossipoff" <mikeo2106 at msn.com> wrote:
From:   "Michael Ossipoff" <mikeo2106 at msn.com>
Subject:    [Proj] (no subject)
Date:   June 27, 2008 12:48:08 PM PDT
To: proj at lists.maptools.org

List members-- 

Last year I defined linearity, as a map property, here, and argued for the desirability of that property on data maps, maps that show d
distributions of such things as temperature, species habitat range, etc. I'd like to improve a little on the definitions that I posted at that time. (by X and Y position, I mean the values of the X and Y map co-ordinates) 

Linearity: 

A map projection is linear if, on maps using that projection, Y position on the map varies linearly with latitude, and X position on the map varies linearly with longitude. 

[end of linearity definition] 

Zonal linearity: 

A map projection is zonally linear if maps using that projection can be divided into regions such that, within each region, Y position varies linearly with latitude, and X position varies linearly with longitude.

[end of zonal linearlity definition] 

I've given linearity the simpler and briefer definition above because all the zonally linear projections, other than my own linear approxmiate (LA) projections, have that briefly defined property that I've defined above as "linearity". So, the only zonally linear maps used anywhere are linear as defined above. My LA projections are the only projections I've heard of that are zonally linear without being linear, by the above definitions. 

In previous postings, I defined an even wordier kind of linearlity, which I'll give a name here, though I don't think this property really needs naming, because I know of no projection that has it without having the zonal linearity that I've defined above. It's alright for its name to be awkward, since I don't expect to say it often: 

Especially-broadly-defined linearity: 

A map is especially-broadly-defined-linear if it can be divided into regions such that within each region, Y position varies linearly with latitude, and if it can also be divided into regions such that X position varies linearly with longitude. 

[end of especially-broadly-defined linearity definition] 

As I said, that property has little value since I know of no projecion possessing it without also possessing zonal linearity as defined above. 

Well, one reason to speak of especially-broadly-defined linearity is to mention that it aparently amounts to the same thing as the linearly interpolable positions property (LIPP) that I defined last year. Since I'm not changing my definition of LIPP, there's no need to repeat it here.

Though not linear, the equidistant conic, Bonne, and Stabius-Werner projections obviously are more convenient for determining and finding lat/lon co-ordinates than are other non-linear projections. They deserve credit for their position-conveniene, and it deserves a name.

Polar linearity:

A map projection is polar-linear if the parallels are concentric circles, and if distance along the each parallel varies linearly with longitude, and if distance along lines passing through the center of the parallels' common center varies linearly with latitude. 

[end of polar linearity definition] 

Of course the equidistant conic, Bonne, and Stabius-Werner are polar-linear. 

Obviously polar linearity isn't as good as linearity, but it's better than what most non-linear prjecions offer for convenience in finding or determining lat/lon co-ordinates.

Some linear projections: 

Cylindrical Equidistant 
Sinusoidal 
Trapezoidal 
Eckhert III (if I remember it correctly) 
Apianus II 

My LA projections, while not being linear as defined here, are zonally linear. 

Mike Ossipoff 



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