[task #15637] Match RA and Dec catalog to X and Y catalog to find WCS

[Natáli Anzanello]

Follow-up Comment #9, task #15637 (project gnuastro):

I think that the matching problem has been solved. Below I explain the steps:

The first thing we needed to fix was the vertices found on each catalog. It's
very important that all the vertices are labeled the same. First, we label the
A and B vertices as the most separated ones. In the query catalog it's just
the Euclidean distance between the points, but on the reference catalog we
have to use the angular distance between the points to get the same vertices.
Prior to that, it was also using the Euclidean distance to the vertices on the
reference catalog, so it would give different most separated A and B for the
two catalogs.
After that, we have to choose the C and D vertices. First we label randomly
the two remaining vertices as C and D and then we compare the ACB and ADB
angles that are less than 180 degrees and choose C to be the one that has the
lesser angle.

Now, we have the A, B, C and D vertices to be the same when dealing with the
same quads and we have to compute their hashes. The hashes were calculated
using Cx = (c1-a1)/(b1-a1), where a1, b1 and c1 are the coordinates along the
axis 1. Now we have the problem related to the rotations: the distance between
the points is the same, but the distance along each axis is not the same! So
the Cx would be different for different axis. The same would happen for Cy, Dx
and Dy.

To solve this, first we transform the celestial coordinates of the reference
catalog into projection plane coordinates (TAN projection) using the midpoint
of AB as the coordinates of the native pole.
We proceed defining new two axis (x and y, where the hashes will be
calculated) using the A-B vector as a 45 degrees line contained in these axis.
Then, we project the C-A and D-A vectors in these axis and get the hashes.

I have tested for a variety of seeds and rotations and it has worked. Let me
know what you think!

Below, a figure showing these steps ('match_overview.png'). I also uploaded a
fix to the figure that I sent in the last post because the indexes were wrong
('vertices_after_match_fixed.png').

(file #51589, file #51590)
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Additional Item Attachment:

File name: vertices_after_match_fixed.png Size:66 KB
   
<https://file.savannah.gnu.org/file/vertices_after_match_fixed.png?file_id=51589>

File name: match_overview.png             Size:132 KB
    <https://file.savannah.gnu.org/file/match_overview.png?file_id=51590>



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