[Proj] proj4 string for perfect sphere

[Bob Fischer]

>>> Although the theory for all this is probably sitting in a geometry book somewhere..
I was referring to this book:
http://www.amazon.com/Computational-Geometry-Surfaces-Performing-Cylinder/dp/1402002025


  Computational Geometry on Surfaces: Performing Computational Geometry
  on the Cylinder, the Sphere, the Torus, and the Cone [Hardcover]

Clara I. Grima 
<http://www.amazon.com/Clara-I.-Grima/e/B001K8V4VK/ref=ntt_athr_dp_pel_1><http://www.amazon.com/Computational-Geometry-Surfaces-Performing-Cylinder/dp/1402002025#> 
(Author), Alberto Márquez 
<http://www.amazon.com/s/ref=ntt_athr_dp_sr_2?_encoding=UTF8&sort=relevancerank&search-alias=books&ie=UTF8&field-author=Alberto%20M%C3%A1rquez> 
(Author)


>> I find this web site very informative for a beginner:
>>
>> http://aa.quae.nl/en/index.html
>> http://aa.quae.nl/en/reken/grootcirkel.html#1
>>
>> this shows all the most basic calculations with examples

Thanks, a lot of nice formulas there.  Unfortunately, it is not entirely 
accurate.  For example, Section 7, "Great Circle on a Topographic Map".  
The Gnomonic Projection is a counter-example to the statement: " It is 
impossible to make a map of the world on which all great circles run 
straight."

> if you need any accuracy and robustness you'll never leave the
> sphere (or ellipsoid). It is much easier to spend a little more time
> with the general spherical calculation than to fix it afterwards for
> 150+ projections for each one with maybe a different method.

This is not always the case.  Let me be specific.

Suppose you have a closed curve on a plane, and you want to know the 
area of it.  Suppose you have described the curve through a 
parameterization, that is functions x(z) and y(z), where z ranges from 0 
to 1.  Then you can use Green's Theorem (Surveyor's Formula) to 
integrate and find the area of this curve.

Now suppose you want to apply this to the sphere.  That is, you have a 
curve on the surface of the sphere described by theata(z) and phi(z).  
All you need to do is choose any old equal area projection.  The 
projection is described as x(theta,phi) and y(theta,phi).  Combine the 
two functions to get a curve on the plane, parameterized by z: 
x(theta(z), phi(z)), y(theta(z), phi(z)).  Now you can use Green's 
Theorem to integrate and find the area of this curve on the plane.  And 
since you chose an area-preserving projection, your answer will be exact.

This is essentially a derivation of Green's Theorem on a curved surface.

Another example where planar geometry can yield exact answers is the 
Cubed Sphere grid.  It uses the Gnomonic projection to create a regular 
grid made up of great circle paths.

> Another trick is to project it to a single
> standard projection which works all over the world and then
> do the plane calculations if they are too hard to be done on the
> sphere.
Unfortunately, there is no such thing.  Every projection has areas of 
the sphere where it fails miserably.  Not just places of high distortion 
(eg, Mercator near the poles), but also places where points that are 
close to each other on the globe end up far apart on the projection (eg, 
Mercator at the International Date Line).

-- Bob

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