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bertin1953.rst: fix formatting
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This fixes the current forward implementation of Peirce Quincuncial proj to correctly flip/reflect out the southern hemisphere to four triangles, and rotate entire result to a square or diamond. (It there resolves the issues identified with pull request https://github.com/OSGeo/PROJ/pull/2230 , where southern hemisphere was wrongly projected over northern, and reverses the restriction to northern hemisphere introduced there). It also adds additional lateral projection of the hemispheres.
- This PR adds an optional parameter `+type` which allows selection of projection. The `+type=square` and `+type=diamond` types match in principle ESRI's twin implementations of square and diamond PQ projs. The **default** if not specified is `+type=diamond`.
- The previous behaviour restricted to the northern hemisphere can be reproduced using the `+type=nhemisphere`, though this is an edge case only.
- An additional `+type=horizontal` and `+type=vertical` rectangular lateral versions have been added that place each hemisphere side-by-side. This is primarily to allow creation of projections such as Greiger Triptychial, which also require the additional optional params `scrollx` or `scrolly` in order to shift parts of the projection from one side of the map to the other.
- Additional documentation has been added to proj description, including quoting the usual meridian used in common usage of projection, and images showing the different types.
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Co-authored-by: Rohit <rohitpingale103@gmail.com>
Co-authored-by: Brendan Jurd <brendan.jurd@geoplex.com.au>
Co-authored-by: Mike Taves <mwtoews@gmail.com>
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- this bumps format_version of tinshift JSON to 1.1 for the new field
fallback_strategy
- the default behaviour without that field is retained
- if fallback_strategy is set to "nearest_side", then points that do not fall
into any of the triangles will be transformed according to the nearest
triangle
- if fallback_centroid is set to "nearest_side", then points that do not fall
into any of the triangles will be transformed according to the triangle
with the nearest centroid
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ortho.rst: fix typo
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#2832)
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necessarily using the operation that appears as first
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Fixes #2638 and fixes #2639
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For https://github.com/OSGeo/PROJ/issues/2645 .
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/ Vanua Levu Grid'
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Closes #2549
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operations' section
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Clarifying defaults for ISEA.
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Typo
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Update Mercator projection, more accurate, faster
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Introduction
------------
The existing formulation for the Mercator projection is
"satisfactory"; it is reasonably accurate. However for a core
projection like Mercator, I think we should strive for full double
precision accuracy.
This commit uses cleaner, more accurate, and faster methods for
computing the forward and inverse projections. These use the
formulation in terms of hyperbolic functions that are manifestly odd
in latitude
psi = asinh(tan(phi)) - e * atanh(e * sin(phi))
(phi = latitude; psi = isometric latitude = Mercator y coordinate).
Contrast this with the existing formulation
psi = log(tan(pi/4 - phi/2))
- e/2 * log((1 + e * sin(phi)) / (1 - e * sin(phi)))
where psi(-phi) isn't exactly equal to -psi(phi) and psi(0) isn't
guaranteed to be 0.
Implementation
--------------
There's no particular issue implementing the forward projection, just
apply the formulas above. The inverse projection is tricky because
there's no closed form solution for the inverse. The existing code
for the inverse uses an iterative method from Snyder. This is the
usual hokey function iteration, and, as usual, the convergence rate is
linear (error reduced by a constant factor on each iteration). This
is OK (just) for low accuracy work. But nowadays, something with
quadratic convergence (e.g., Newton's method, number of correct digits
doubles on each iteration) is preferred (and used here). More on this
later.
The solution for phi(psi) I use is described in my TM paper and I
lifted the specific formulation from GeographicLib's Math::tauf, which
uses the same underlying machinery for all conformal projections. It
solves for tan(phi) in terms of sinh(psi) which as a near identity
mapping is ideal for Newton's method.
For comparison I also look at the approach adopted by Poder + Engsager
in their TM paper and implemented in etmerc. This uses trigonometric
series (accurate to n^6) to convert phi <-> chi. psi is then given by
psi = asinh(tan(chi))
Accuracy
--------
I tested just the routines for transforming phi <-> psi from merc.cpp
and measured the errors (converted to true nm = nanometers) for the
forward and inverse mapping. I also included in my analysis the
method used by etmerc. This uses a trigonometric series to convert
phi <-> chi = atan(sinh(psi)), the conformal latitude.
forward inverse
max rms max rms
old merc 3.60 0.85 2189.47 264.81
etmerc 1.82 0.38 1.42 0.37
new merc 1.83 0.30 2.12 0.31
1 nm is pretty much the absolute limit for accuracy in double
precision (1 nm = 10e6 m / 2^53, approximately), and 5 nm is probably
the limit on what you should routinely expect. So the old merc
inverse is considerably less accurate that it could be. The old merc
forward is OK on accuracy -- except that if does not preserve the
parity of the projection.
The accuracy of etmerc is fine (the truncation error of the 6th order
series is small compared with the round-off error). However,
situation reverses as the flattening is increased. E.g., at f =
1/150, the max error for the inverse projection is 8 nm. etmerc is OK
for terrestrial applications, but couldn't be used for Mars.
Timing
------
Here's what I get with g++ -O3 on various Linux machines with recent
versions of g++. As always, you should take these with a grain of
salt. You might expect the relative timings to vary by 20% or so when
switching between compilers/machines. Times per call in ns =
nanoseconds.
forward inverse
old merc 121 360
etmerc 4e-6 1.4
new merc 20 346
The new merc method is 6 times faster at the forward projection and
modestly faster at the inverse projection (despite being more
accurate). The latter result is because it only take 2 iterations of
Newton's method to get full accuracy compared with an average of 5
iterations for the old method to get only um accuracy.
A shocking aspect of these timings is how fast etmerc is. Another is
that forward etmerc is streaks faster that inverse etmerc (it made be
doubt my timing code). Evidently, asinh(tan(chi)) is a lot faster to
compute than atan(sinh(psi)). The hesitation about adopting etmerc
then comes down to:
* the likelihood that Mercator may be used for non-terrestrial
bodies;
* the question of whether the timing benefits for the etmerc method
would be noticeable in a realistic application;
* need to duplicate the machinery for evaluating the coefficients
for the series and for Clenshaw summation in the current code
layout.
Ripple effects
==============
The Mercator routines used the the Snyder method, pj_tsfn and pj_phi2,
are used in other projections. These relate phi to t = exp(-psi) (a
rather bizarre choice in my book). I've retrofitted these to use the
more accurate methods. These do the "right thing" for phi in [-pi/2,
pi/2] , t in [0, inf], and e in [0, 1). NANs are properly handled.
Of course, phi = pi/2 in double precision is actually less than pi/2,
so cos(pi/2) > 0. So no special handling is needed for pi/2. Even if
angles were handled in such a way that 90deg were exactly represented,
these routines would still "work", with, e.g., tan(pi/2) -> inf.
(A caution: with long doubles = a 64-bit fraction, we have cos(pi/2) <
0; and now we would need to be careful.)
As a consequence, there no need for error handling in pj_tsfn; the
HUGE_VAL return has gone and, of course, HUGE_VAL is a perfectly legal
input to tsfn's inverse, phi2, which would return -pi/2. This "error
handling" was only needed for e = 1, a case which is filtered out
upstream. I will note that bad argument handling is much more natural
using NAN instead of HUGE_VAL. See issue #2376
I've renamed the error condition for non-convergence of the inverse
projection from "non-convergent inverse phi2" to "non-convergent
sinh(psi) to tan(phi)".
Now that pj_tsfn and pj_phi2 now return "better" results, there were
some malfunctions in the projections that called them, specifically
gstmerc, lcc, and tobmerc.
* gstmerc invoked pj_tsfn(phi, sinphi, e) with a value of sinphi
that wasn't equal to sin(phi). Disaster followed. I fixed this.
I also replaced numerous occurrences of "-1.0 * x" by "-x".
(Defining a function with arguments phi and sinphi is asking for
trouble.)
* lcc incorrectly thinks that the projection isn't defined for
standard latitude = +/- 90d. This happens to be false (it reduces
to polar stereographic in this limit). The check was whether
tsfn(phi) = 0 (which only tested for the north pole not the south
pole). However since tsfn(pi/2) now (correctly) returns a nonzero
result, this test fails. I now just test for |phi| = pi/2. This
is clearer and catches both poles (I'm assuming that the current
implementation will probably fail in these cases).
* tobmerc similarly thinks that phi close to +/- pi/2 can't be
transformed even though psi(pi/2) is only 38. I'm disincline to
fight this. However I did tighten up the failure condition
(strict equality of |phi| == pi/2).
OTHER STUFF
===========
Testing
-------
builtins.gei: I tightened up the tests for merc (and while I was about
it etmerc and tmerc) to reflect full double precision accuracy. My
test values are generated with MPFR enabled code and so should be
accurate to all digits given. For the record, for GRS80 I use f =
1/298.2572221008827112431628366 in these calculations.
pj_phi2_test: many of the tests were bogus testing irrelevant input
parameters, like negative values of exp(-psi), and freezing in the
arbitrary behavior of phi2. I've reworked most for the tests to be
semi-useful. @schwehr can you review.
Documentation
-------------
I've updated merc.rst to outline the calculation of the inverse
projection.
phi2.cpp includes detailed notes about applying Newton's method to
find tan(phi) in terms of sinh(psi).
Future work
-----------
lcc needs some tender loving care. It can easily (and should) be
modified to allow stdlat = +/- 90 (reduces to polar stereographic),
stdlat = 0 and stdlat_1 + stdlat_2 = 0 (reduces to Mercator). A
little more elbow grease will allow the treatment of stdlat_1 close to
stdlat_2 using divided differences. (See my implementation of the
LambertConformalConic class in GeographicLib.)
All the places where pj_tsfn and pj_phi2 are called need to be
reworked to cut out the use of Snyder's t = exp(-psi() variable and
instead use sinh(psi).
Maybe include the machinery for series conversions between all
auxiliary latitudes as "support functions". Then etmerc could use
this (as could mlfn for computing meridional distance). merc could
offer the etmerc style projection via chi as an option when the
flattening is sufficiently small.
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by a number of projected CRS in Colombia (fixes #589)
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Also define merc (resp. tmerc) as the conformal projection in which the
equator (resp. a chosen meridan} projects to a straight line at constant
scale.
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This addresses item 1 in issue #2387
Things to note:
* I made "editorial" changes to the text. The virtues and vices of
Mercator are a hot topic. So check these out. (I judged that the
text I replaced to be pretty misleading.)
* I include the radius of the sphere/ellipsoid in the formulas (and I
did this also for my mods for tmerc documentation). Surely this is
better than leaving the reader to figure out how this is introduced.
* I include the "old-style" (ca 18th century) formulas and the newer
ones in terms of hyperbolic functions. The former may be the familiar
ones, but the latter are better for computation (more succinct, more
accurate, faster, preserve parity).
* For the inverse ellipsoidal transformation, I just say that the
formula for psi is inverted iteratively. This is probably sufficient,
but it could be expanded later.
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* Document the default poder_engsager algorithm for tmerc.
* Give exact expression for phi' in terms of phi
* Aadd another datapoint on range of applicability + explanation for
complex numbers.
* Update tmerc figure with one reflecting poder/engsager algo.
Courtesy of @hobu.
Co-authored-by: Charles Karney <charles.karney@sri.com>
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* implement conda package building
* paths
* need libtool
* PLATFORM check
* point to my PROJ feedstock for now
* point to PROJ repos
* plot building and artifact upload
* syntax
* add proj conda package for plotting
* retab
* no doxygen
* syntax
* update docs Makefile
* doc building
* needs
* consolidate doc/plots
* plot updates
* put updated plot output into docs
* register spelling module
* use v2 download-artifact
* artifact upload
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Implement ellipsoidal formulation of +proj=ortho (fixes #397)
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Implements RFC-6
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- Map ESRI 'Local' to +proj=ortho when Scale_Factor = 1 and Azimuth = 0
- Map ESRI 'Orthographic' to a PROJ WKT2 'Orthographic (Spherical)'
which maps to +proj=ortho +f=0 to froce spherical evaluation
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+lat_0 and +k_0 (#2255)
Refs #2254
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- Move the generic method initiated from adams_ws2 to a
pj_generic_inverse_2d() method
- Use it in adams_ws2
- Use it in wink2
Fixes https://github.com/qgis/QGIS/issues/35512
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Limit peirce_q to northern hemisphere, and fix images for adams_hemi, guyou and peirce_q
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- Central lon for zone 2 should be -d10, not d10
- Extra lobe was missing for zone 11
- New figure generated
- New test suite values generated
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- The current implementation of the Interrupted Goode Homolosine
projection emphasizes land area. This is a compliment projection
that emphasizes ocean area.
- A value of lon0=-160 produces a reasonable real-world map.
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Add a +proj=defmodel transformation for multi-component time-based deformation models
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deformation models
Fixes #1001
Co-authored-by: Chris Crook <ccrook@linz.govt.nz>
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regenerate associated images
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