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Do you have a programmatic version of the algorithm? It's something
that would be really useful for a global map provider. I am working
with the Royal Tropical Institute here in Amsterdam (<a class="moz-txt-link-abbreviated" href="http://www.kit.nl">www.kit.nl</a>), on an
index for their 10.000s of historical maps from all over the globe. In
the long run we would like to offer some thematic functionality, with
an optimal projection, of course <br>
<br>
Jan<br>
<br>
On 16-2-2010 4:55, strebe wrote:
<blockquote cite="mid:12E44B3B.8DB7.4AB8.B9B9.3D0EE3F96D84@aol.com"
type="cite">
<div><br>
</div>
<div>Jan & colleagues:</div>
<div><br>
</div>
<div>Regarding the question of optimal standard parallels for Albers,
I found my notes from 8 years ago. To summarize, there is no good rule,
so perhaps the exact calculation I describe below has more importance
than just academic.</div>
<div><br>
</div>
<div>Regards,</div>
<div>— daan Strebe</div>
<div><br>
</div>
<div>
<div><font class="Apple-style-span" face="monospace">________________________</font></div>
<div><font class="Apple-style-span" face="monospace"><br>
</font></div>
<div><font class="Apple-style-span" face="monospace"><br>
</font></div>
</div>
<div><span class="Apple-style-span" style="font-family: monospace;">The
problem: given two latitudinal extents phi[N] and phi[S], find the
optimal standard parallels for the Albers projection. Optimal is
defined to mean that the maximum angular deformation between the two
standard parallels equals the maximum angular deformation at the two
latitudinal extents. In this way one is assured that the maximum
deviation from true remains minimal throughout the region of interest.<br>
<br>
In a breathtakingly brief treatment, Melluish touches on the issue by
describing how to select standard parallels such that the "error" on
the extreme parallels is equal in magnitude and opposite in sign to the
error of the central parallel. The distortion measurement is equivalent
to mine, but the central parallel (the average of the north and south
extreme parallels) generally does not carry the greatest distortion, so
his method does not yield an optimal map. Hinks discusses the problem
precisely, even noting that the central parallel is not the parallel of
greatest distortion... but his treatment applies to the equidistant
conic, not the equal-area, contrary to Snyder's comment in USGS P.P
1395.<br>
<br>
We define a family of Albers projections to be all Albers projections
characterized by a common 'central' latitude at which angular
deformation reaches a peak between the two standard parallels, or sinks
to zero in the case of one standard parallel. I have placed 'central'
between quote marks because it is not any simple average of the
standard parallels. We can demonstrate that the ratio of the angular
deformation of any two points on the map remains constant for every
member of the family. This is the beautiful and useful characteristic
of the family. The property arises out of a happy mathematical
coincidence.<br>
<br>
First I declare the member of the family with only one standard
parallel to be the canonical form. Because of the family's property of
invariant scale ratios, we can first reduce the problem of choosing an
optimal Albers projection to the single standard parallel case, which
is much more tractable than trying to deal with two free parameters.
This gives us a projection with zero distortion along the canonical
parallel and identical distortion at the northern and southern
extremities. Once we have found the canonical projection, we can choose
the member of its family that produces the same magnitude of distortion
at the canonical parallel as at the extreme parallels. We are
guaranteed that the projection that fulfills this condition is one from
the family of the canonical projection.<br>
<br>
Let phi[S] denote the southern parallel of interest; phi[N] the
northern parallel of interest; phi[0] the canonical parallel; phi[1] a
standard parallel; and phi[2] the other standard parallel.<br>
<br>
phi[S] and phi[N] are given. From them we must determine phi[0],
phi[1], and phi[2], such that the magnitude of the angular deformation
at phi[0] is the same as the magnitude of the angular deformation at
phi[N] and phi[S]. Furthermore, there must be no greater angular
deformation registered anywhere between the phi[S] and phi[N]. Given
those conditions:<br>
<br>
1) sin (phi[0]) = A +/- sqrt (A^2 - 1)<br>
<br>
where<br>
<br>
2) A = (sin (phi[N]) * cos^2 (phi[S]) - sin (phi[S]) * cos^2 (phi[N]))
/ (cos^2 (phi[S] - cos^2 (phi[N]))<br>
<br>
That establishes the canonical parallel of the canonical form, and
therefore the family, wherein lies the solution.<br>
<br>
If we arbitrarily select some phi[1], one may ask how to arrive at the
correct phi[2] within the family, given phi[0].<br>
<br>
3) sin (phi[2]) = [sin (phi[1]) - 2 sin (phi[N]) + sin (phi[1]) sin^2
(phi[N])] / [2 sin (phi[N]) * sin (phi[1]) - sin^2 (phi[N]) - 1]<br>
<br>
However, we do not yet know the proper phi[1]. With much manipulation,
we discover the iterative equation:<br>
<br>
4) cos^2 (phi[1]) * sqrt (1 + sin^2 (phi[0]) - 2 sin (phi[0]) * sin
(phi[S])) - cos (phi[S]) (1 + sin^2 (phi[0] - 2 sin (phi[0]) sin
(phi[1]) = 0<br>
<br>
We solve for phi[1] with (4) and then use (3) to compute phi[2]. Now
have the optimal Albers for any circumstance.<br>
<br>
The literature fixates quite a bit on rules of thumb for choosing the
standard parallels, with some texts recommending inward 1/5th the
meridional separation of the extreme parallels. Other texts recommend
1/7th, and yet others recommend 1/6th. My own calculations based on the
above criteria show that the correct values range from 5% to 25% for
reasonable conics, and the symmetry between north and south is not
necessarily very good. Hence, no rule of thumb is very good. The higher
the latitude of the northern parallel, the greater the asymmetry. The
asymmetry remains remarkable even if both latitudes are high. For
instance, extreme latitudes at 50° and 70° result in standard parallels
placed optimally at roughly 10% below phi[N] and 18% above phi[S].<br>
<br>
Deetz's and Adams's recommendation of 29°30' and 45°30' for the US are
pretty good, of course. Given maximum extents of 24°34' and 49°25', the
optimal standard parallels are 28°42' and 46°19'. If we just can't
bring ourselves to treat Lake of the Woods equitably then the upper
standard parallel drops to 45°56'. The lower parallel only drops by 5'.
The former map yield a maximum east-west scale distortion of 1.19%; the
latter yields 1.15% (excluding Lake of the Woods, of course). Deetz's
and Adams's recommendation yields 0.98% in the heartland; 1.25% at the
northern border; 1.45% at the northern tip of the Lake of the Woods
boundary; and 1.37% at Key West.<br>
</span></div>
<div><font class="Apple-style-span" face="monospace"><br>
</font></div>
<div><font class="Apple-style-span" face="monospace">________________________</font></div>
<div><br>
</div>
<br>
On Feb 15, 2010, at 1:05:19 PM, "Jan Hartmann"
<a class="moz-txt-link-rfc2396E" href="mailto:j.l.h.hartmann@uva.nl"><j.l.h.hartmann@uva.nl></a> wrote:<br>
<blockquote
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width="70px"><span>From:</span></td>
<td
style="font-family: 'Lucida Grande'; font-size: 8pt; color: black; text-align: left; vertical-align: top; padding-left: 5px;"><span
title=""Jan Hartmann" <j.l.h.hartmann@uva.nl>">"Jan Hartmann"
<a class="moz-txt-link-rfc2396E" href="mailto:j.l.h.hartmann@uva.nl"><j.l.h.hartmann@uva.nl></a></span></td>
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<td
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width="70px"><span>Subject:</span></td>
<td
style="font-family: 'Lucida Grande'; font-size: 8pt; color: black; text-align: left; vertical-align: top; padding-left: 5px;"><span
style="font-weight: bold;">Re: [Proj] Standard projection for
Mediterranean basin</span></td>
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<td
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width="70px"><span>Date:</span></td>
<td
style="font-family: 'Lucida Grande'; font-size: 8pt; color: black; text-align: left; vertical-align: top; padding-left: 5px;"><span>February
15, 2010 1:05:19 PM PST</span></td>
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width="70px"><span>To:</span></td>
<td
style="font-family: 'Lucida Grande'; font-size: 8pt; color: black; text-align: left; vertical-align: top; padding-left: 5px;"><span
title=""PROJ.4 and general Projections Discussions" <proj@lists.maptools.org>">"PROJ.4
and general Projections Discussions" <a class="moz-txt-link-rfc2396E" href="mailto:proj@lists.maptools.org"><proj@lists.maptools.org></a></span></td>
</tr>
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<td
style="font-family: 'Lucida Grande'; font-size: 8pt; color: gray; text-align: right; vertical-align: top; font-weight: bold;"
width="70px"><span>Cc:</span></td>
<td
style="font-family: 'Lucida Grande'; font-size: 8pt; color: black; text-align: left; vertical-align: top; padding-left: 5px;"><span
title="strebe <strebe@aol.com>">strebe <a class="moz-txt-link-rfc2396E" href="mailto:strebe@aol.com"><strebe@aol.com></a></span></td>
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<div id="felix-mail-content-block"
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<div>What's the 1/6 rule, Daan, and how much does it deviate from
the optimal formula?<span class="Apple-converted-space"> </span><br>
<br>
BTW Nice Java applet to study projection deformations here:<br>
<br>
<a moz-do-not-send="true" class="moz-txt-link-freetext"
href="http://www.uff.br/mapprojections/Albers_en.html"
title="http://www.uff.br/mapprojections/Albers_en.html">http://www.uff.br/mapprojections/Albers_en.html</a><br>
<br>
Pity they didn't make it possible to alter projection parameters like
standard parallels.<br>
<br>
Jan<span class="Apple-converted-space"> </span><br>
<br>
On 15-2-2010 21:04, strebe wrote:
<blockquote cite="mid:2C1B7FAB.76DF.4D31.AEC0.3B87D1ABB14C@aol.com"
type="cite">
<div><br>
</div>
<div>I do not understand Dr. Mugnier's response. Using the
authalic latitude does not yield an equal-area projection if you
project using a conformal projection. I cannot think of any use for
mixing authalic latitude with conformal projection.</div>
<div><br>
</div>
<div>The purpose of the authalic latitude is to provide an
equal-area projection of the ellipsoid onto the sphere. From there you
may project to the plane using any spherical equal-area projection,
yielding an equal-area projection of the ellipsoid onto the plane.
While this technique will not preserve other properties of the
spherical projection (such as standard parallels or constant scale
along some meridian or somesuch), at least it drastically increases the
available projections for ellipsoidal use.</div>
<div><br>
</div>
<div>I have, by the way, formulæ somewhere that I derived for
computing the optimal standard parallels for Albers given the
north/south extents of the map. I will post them if I happen to find
them. The 1/6th rule would be fine for any real use, of course. The
optimal is more of academic interest.</div>
<div>
<div><br>
</div>
</div>
<div>Regards,</div>
<div>
<div>— daan Strebe</div>
<div><br>
</div>
</div>
</blockquote>
</div>
</div>
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</span></blockquote>
<br>
<div><br>
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