Talk:Mollweide projection

message: Is there a wiki page on interrupted Mollweide projection maps?  See, for example, this page at Clarkson University, or this one at Colorado State.  There are also some royalty-free maps available here (though I doubt they'd let those go under CopyLeft).
NASA has some software available here.  They have an interrupted Mollweide map of the oceans.  I s'pose the code could be adapted to produce maps of land masses instead.  Note that these are produced by NASA (and so are PD).
Anyone wanna add a section to this article?  And add some images?  Maybe I'll get bored later this week and add it myself.  Sigh.
cheers, 66.25.169.151 (talk) 07:58, 22 November 2009 (UTC)
thread_title:  Interrupted Mollweide? 

message: The second line in second paragraph reads "It was popularized by Jacques Babinet in 1857, giving it the name homalographic projection.[1]" <-- Should it be homolographic with an "O" vs. "A". Wasn't sure if this  is an old spelling or not. Please confirm.
Cheers  —Preceding unsigned comment added by 71.228.98.160 (talk) 19:15, 17 January 2011 (UTC)
replies: text: Both spellings are in use. “Homalographic” was the name Babinet applied, but the projection was used in many 19th century star atlases, for example, where the variations in spelling cropped up. Strebe (talk) 21:21, 17 January 2011 (UTC)
thread_title:  HomAlographic 

message: The sqrt(2) linear scaling factor can be removed from expressions for both x and y to simplify them.
This mapping has also very simple inverse formula, not needing any iteration - first solve theta from y, then solve latitude from theta equation and longitude from equation for x.
62.24.73.63 (talk) 20:45, 6 August 2011 (UTC)
replies: text: The scale factor is to promote a reasonable nominal scale for the map, which is normal practice in the cartographic literature. The inverse is simple; feel free to add it. Strebe (talk) 21:30, 6 August 2011 (UTC)
replies: text:  Strebe, I do not understand. The way it is written right now, the x-coordinates run from -2√2 to +2√2, and the y-coordinates run from -√2 to +√2. Wouldn’t it make sense to get rid of the extraneous √2 and have x run from -2 to 2, and y from -1 to 1? I am going to make this change AND make a note in the text of the article indicating the range of x and y, because right now there is no such information. Qaanol (talk) 19:33, 31 August 2014 (UTC)
replies: text: Your edit contradicts the footnote source, which you should not do. Please discuss these things further if you don’t understand rather than encoding your misunderstanding in the article. The √2 factor is not “extraneous”. Its presence yields a mapped space having the same area as the surface area of the generating globe, which is, after all, one of the goals of an equal-area map. Strebe (talk) 20:47, 31 August 2014 (UTC)

text: Ah, that makes sense: the map is made to have the same area as the globe it is projected from. Thank you. I’ve added a few words to note the range of x and y. Qaanol (talk) 22:46, 31 August 2014 (UTC)
thread_title:  Scaling factor and inverse formula 

message: I will be replacing images on the various map projection pages. Presently many are on a satellite composite image from NASA that, while realistic, poorly demonstrates the projections because of dark color and low contrast. I have created a stylization of the same data with much brighter water areas and a light graticule to contrast. See the thumbnail of the example from another article. Some images on some pages are acceptable but differ stylistically from most articles; I will replace these also.
The images will be high resolution and antialiased, with 15° graticules for world projections, red, translucent equator, red tropics, and blue polar circles.
Please discuss agreement or objections over here (not this page). I intend to start these replacements on 13 August. Thank you.
Strebe (talk) 22:45, 6 August 2011 (UTC)
thread_title: Replace low-contrast images

message: Hello! This is a note to let the editors of this article know that :File:Mollweide projection SW.jpg will be appearing as picture of the day on January 2, 2017. You can view and edit the POTD blurb at Template:POTD/2017-01-02. If this article needs any attention or maintenance, it would be preferable if that could be done before its appearance on the Main Page. — Chris Woodrich (talk) 08:49, 22 December 2016 (UTC)
replies: text: Perfect, Chris Woodrich! Strebe (talk) 23:15, 22 December 2016 (UTC)
thread_title:  [[:File:Mollweide projection SW.jpg]] to appear as POTD soon 

message: === Newton-Raphson with fast convergence everywhere ===
I don't have a citation so I didn't modify the article, but the issue of slow convergence near the poles can be solved easily. I'm putting this here for future reference or if it is useful to someone.
Modifying the Newton-Raphson scheme to use a different \theta_0 allows fast convergence for every \varphi\in[-\pi/2,\pi/2]:
:\begin{align} \theta_0 &= \left(\frac{\pi}{2} - \sqrt[3]{\frac{3\pi}{8} \left(\frac{\pi}{2} - |\varphi|\right)^2}\right) \sgn \varphi, \\ \theta_{n+1} &= \theta_n - \frac{2 \theta_n + \sin 2\theta_n - \pi \sin \varphi}{2 + 2 \cos 2\theta_n}.\end{align}
For infinite-precision arithmetic, the absolute error is just under 2\cdot10^{-12} after 3 iterations and 2\cdot10^{-24} after 4 iterations.
For double-precision arithmetic, Newton-Raphson iterations will decrease precision when |\varphi| > 1.570762, so you should use \theta_0 in that case, and \theta_3 otherwise. The resulting error is under 2\cdot10^{-10}.
--Mydartvilla (talk) 16:52, 1 May 2019 (UTC)
thread_title:  Solving precision issues near the poles 

message: I don't have a citation so I didn't modify the article, but the issue of slow convergence near the poles can be solved easily. I'm putting this here for future reference or if it is useful to someone.
Modifying the Newton-Raphson scheme to use a different \theta_0 allows fast convergence for every \varphi\in[-\pi/2,\pi/2]:
:\begin{align} \theta_0 &= \left(\frac{\pi}{2} - \sqrt[3]{\frac{3\pi}{8} \left(\frac{\pi}{2} - |\varphi|\right)^2}\right) \sgn \varphi, \\ \theta_{n+1} &= \theta_n - \frac{2 \theta_n + \sin 2\theta_n - \pi \sin \varphi}{2 + 2 \cos 2\theta_n}.\end{align}
For infinite-precision arithmetic, the absolute error is just under 2\cdot10^{-12} after 3 iterations and 2\cdot10^{-24} after 4 iterations.
For double-precision arithmetic, Newton-Raphson iterations will decrease precision when |\varphi| > 1.570762, so you should use \theta_0 in that case, and \theta_3 otherwise. The resulting error is under 2\cdot10^{-10}.
--Mydartvilla (talk) 16:52, 1 May 2019 (UTC)
thread_title:  Newton-Raphson with fast convergence everywhere 

message: The inverse transform in double-precision arithmetic should also use the formula
\varphi = \begin{cases}\arcsin \frac{2 \theta + \sin 2\theta}{\pi} & \text{when }|\theta| < \pi/2 - 10^{-3} \\ \left(\frac{\pi}{2} - \sqrt{\frac{8}{3\pi} \left(\frac{\pi}{2} - |\theta|\right)^3}\right) \sgn \theta & \text{otherwise} \end{cases}
Combined with the Newton-Raphson modification, the error when converting back and forth is under 3\cdot10^{-10} in double precision, instead of more than 6\cdot10^{-6} when only Newton-Raphson is modified. --Mydartvilla (talk) 17:53, 1 May 2019 (UTC)
thread_title:  Accurate inverse transform