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cite="mid:25548613.8594601334741938692.JavaMail.support.mn@elisanet.fi"
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<pre wrap="">Although the theory for all this is probably sitting in a geometry book somewhere..</pre>
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I was referring to this book:<br>
<a class="moz-txt-link-freetext" href="http://www.amazon.com/Computational-Geometry-Surfaces-Performing-Cylinder/dp/1402002025">http://www.amazon.com/Computational-Geometry-Surfaces-Performing-Cylinder/dp/1402002025</a><br>
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<h1 class="parseasinTitle "><small><small><small>
<span id="btAsinTitle">Computational Geometry on Surfaces:
Performing Computational Geometry on the Cylinder, the
Sphere, the Torus, and the Cone <span
style="text-transform: capitalize; font-size: 16px;">[Hardcover]</span></span>
</small></small></small></h1>
<span>
<span class="contributorNameTrigger"><a asin="B001K8V4VK"
id="contributorNameTriggerB001K8V4VK"
href="http://www.amazon.com/Clara-I.-Grima/e/B001K8V4VK/ref=ntt_athr_dp_pel_1">Clara
I. Grima</a><a asin="B001K8V4VK"
href="http://www.amazon.com/Computational-Geometry-Surfaces-Performing-Cylinder/dp/1402002025#"><span
class="contributorChevron" style="margin-left:5px;"><span
class="swSprite s_chevron"></span></span></a></span>
(Author), <a
href="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">Alberto
Márquez</a> (Author)
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<blockquote
cite="mid:25548613.8594601334741938692.JavaMail.support.mn@elisanet.fi"
type="cite">
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<pre wrap="">
I find this web site very informative for a beginner:
<a class="moz-txt-link-freetext" href="http://aa.quae.nl/en/index.html">http://aa.quae.nl/en/index.html</a>
<a class="moz-txt-link-freetext" href="http://aa.quae.nl/en/reken/grootcirkel.html#1">http://aa.quae.nl/en/reken/grootcirkel.html#1</a>
this shows all the most basic calculations with examples</pre>
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<br>
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: "
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It is impossible to make a map of the world on which all great
circles
run straight."<br>
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<blockquote
cite="mid:6420073.8592851334740825515.JavaMail.support.mn@elisanet.fi"
type="cite">
<pre wrap="">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.</pre>
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<br>
This is not always the case. Let me be specific.<br>
<br>
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.<br>
<br>
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.<br>
<br>
This is essentially a derivation of Green's Theorem on a curved
surface.<br>
<br>
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.<br>
<br>
<blockquote
cite="mid:6420073.8592851334740825515.JavaMail.support.mn@elisanet.fi"
type="cite">
<pre wrap="">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.</pre>
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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).<br>
<br>
-- Bob<br>
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