Talk:Peirce quincuncial projection

Image

It would be great to have a better image that shows meridians and main parallels. It will make it clearer how the Equator maps to a square.Dmgerman 17:13, 4 July 2007 (UTC)

Stereographic?

In this edit it was asserted that this projection is not based in any meaningful way on the stereographic projection. I would agree that the unqualified and unexplained assertion that it's "based on the stereographic projection" is not very informative and probably incomprehensible to anyone who hasn't thought about the math.

However: Jacobi's elliptic functions take the torus to the plane, and then the (inverse of the) stereographic projection takes the plane to the Riemann sphere. Hence the correspondence between the torus and the sphere is a composition of the two. You'll see this discussed if you read some of the referenced math articles. Michael Hardy (talk) 20:51, 24 October 2010 (UTC)

That’s true, but not important. I could say the same thing about the ellipsoidal transverse Mercator. Really, any conformal projection can be shuffled through the stereographic before reaching its “destination”, since the stereographic is a convenient way of transforming the sphere to the plane. From there what happens is nothing but function composition. But the stereographic is not necessary; other functions will do. Any conformal transformation to the plane will do. Strebe (talk) 03:50, 25 October 2010 (UTC)

@Strebe: You say other functions will do, but then the next function with which it would be composed would not be the inverse of the standard Jacobi elliptic function. Michael Hardy (talk) 01:46, 23 December 2016 (UTC)

Of course that is true. What is the significance? Strebe (talk) 21:16, 24 December 2016 (UTC)

Replace low-contrast images

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) 22:46, 6 August 2011 (UTC)

Meridian

For the illustration, I think it would be better to choose a prime meridian that avoids Africa. That is the way it's presented here: http://www.quadibloc.com/maps/mcf0703.htm

I have no political agenda, I just think it looks bad right now! Alright, so I have an aesthetic agenda. Kyle Cronan (talk) 03:55, 20 March 2012 (UTC)

Done. Strebe (talk) 04:03, 23 July 2012 (UTC)

Awesome, love it! Kyle Cronan (talk) 00:09, 7 August 2012 (UTC)

File:Peirce quincuncial projection SW 20W.JPG to appear as POTD soon

Hello! This is a note to let the editors of this article know that File:Peirce quincuncial projection SW 20W.JPG will be appearing as picture of the day on April 14, 2016. You can view and edit the POTD blurb at Template:POTD/2016-04-14. 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) 23:41, 27 March 2016 (UTC)

Picture of the day

The Peirce quincuncial projection is a conformal map projection developed by Charles Sanders Peirce in 1879, while he was working at the U.S. Coast and Geodetic Survey. In the normal aspect Peirce's projection presents the Northern Hemisphere in a square; the Southern Hemisphere is split into four 90°–45°–45° triangles surrounding it so that the whole map forms a larger square.Map: Strebe, using Geocart

Hello Chris Woodrich, and thanks for your tireless efforts! I pared back mention of Schwarz-Christoffel transformation from the image description under the theory that only mathematicians would benefit from that part of the blurb. If anyone thinks otherwise, let's talk. Thanks. Strebe (talk) 17:59, 28 March 2016 (UTC)

• Alright.  — Chris Woodrich (talk) 00:29, 29 March 2016 (UTC)

Formal description

I am very suspicious of "tan (p/2)" in the "Formal description" section. Maybe arctan should be there. I will check this out and return. Michael Hardy (talk) 01:20, 23 December 2016 (UTC)

Now I am more than suspicious: What was there was right, but described so badly as to be wrong. In this version, the page said:

A point P on the Earth's surface, a distance p from the North Pole with longitude θ and latitude λ is first mapped to a point (p, θ) of the plane through the equator, viewed as the complex plane with coordinate w; this w coordinate is then mapped to another point (x, y) of the complex plane (given the coordinate z) by an elliptic function of the first kind. Using Gudermann's notation for Jacobi's elliptic functions, the relationships are

$\tan \left({\frac {p}{2}}\right)e^{i\theta }=\mathrm {cn} \left(z,{\frac {1}{\sqrt {2}}}\right),{\mbox{ where }}w=pe^{i\theta }{\mbox{ and }}z=x+iy.$

First the point is given the name (capital) P, but that name is never used after that, so it serves no purpose. Then it is said to be at a distance (lower-case) p from the north pole, which ultimately makes sense if that is taken to be angular distance. Then it says "latitude λ" but never again refers to λ, so that also serves no purpose. One would have λ = ([right angle] − p), so we have two notations for latitude, and only one of them is subsequently used. Then it says it's mapped to a point (p, θ) in the plane through the equator. That is grossly wrong. It should have said it's mapped to a point ((tan(p/2)), θ) in that plane, where the components of that pair are the polar coordinates. That is the standard stereographic projection. Finally, let us note that if

$\tan {\frac {p}{2}}e^{i\theta }=\operatorname {cn} (z),$

then

$z=\operatorname {cn} ^{-1}\left(\tan {\frac {p}{2}}e^{i\theta }\right),$

where cn⁻¹ is the multiple-valued inverse of the elliptic function. Its multiple-valued nature is seen in the fact that each point on the surface of the earth is mapped to infinitely many points in the plane in this doubly periodic projection.

This explains both my latest edits and the reason why they took so long, i.e. they took a while because of the cryptic nature of the way the article was written, especially in saying (p, θ) where it needed to say ((tan(p/2)), θ). Michael Hardy (talk) 23:41, 25 December 2016 (UTC)

@Strebe: : Your edit summary suggests you should look at what I wrote about the multiple-valued inverse above. Michael Hardy (talk) 23:42, 25 December 2016 (UTC)

@Michael Hardy: I think we just need to start over completely. It’s now clearly wrong. The elliptic integral of the first kind is the inverse of the elliptic function cn, so the text you changed to cannot be right. The page, substantially in its present form, seems to have sprung up ab initio in 2007 from User:Dmgerman’s start. I can’t find a reference in the projection literature that renders the projection in the form that “he” gave, though possibly there are mathematical texts that do so. I could try to derive what he has from forms given in sources I have, but ultimately that would be original research. Rather than laboring over what, ultimately, is hearsay, we should render the math directly as found in a reliable source.

We have two primary forms. Ultimately they are mathematically equivalent, of course, but they spring from very different conceptual models. The form given by Peirce himself (1877) uses elliptic integrals of the first kind and avoids complex functions. L.P. Lee (1976) has this to say about the general system as described by Guyou (1887) and followed by Adams (1925): Adams used the mathematically ingenious but laborious method, first described by Guyou, involving elliptic coordinates on the sphere and the use of tables of elliptic integrals. The elliptic coordinates do not form an isometric system, although isometric coordinates can be derived from them. Hence Adams’s reference to elliptic isometric coordinates is misleading. The form that Lee propounds uses elliptic functions like what’s in the article now, and is quite pithy. It’s plausibly equivalent, but I don’t see the point of establishing that. We should just replace what’s there now. Strebe (talk) 07:16, 26 December 2016 (UTC)

I think a comparison with the TOAST projection might be useful

https://docs.worldwidetelescope.org/data-guide/1/spherical-projections/toast-projection/

I had at first naively assumed they were the same, until I found out there was an specific name for the wrong guess I had made and found the image of the Earth with that projection does indeed significantly differ from how it looks with the Quincuncial projection. I understand the link above may perhaps not be sufficient for Wikipedia guidelines, but it might be good start for someone with more refined wikipedian skills than me.

Oh, and I also found a paper that goes bit more into depth into this projection and some variants, that might also be helpful: Octahedron-based Projections as Intermediate Representations for Computer Imaging: TOAST, TEA, and More

-- 95.92.227.32 (talk) 16:46, 4 October 2024 (UTC)

I have never heard of the "TOAST projection", but your link is about taking a hierarchical triangular mesh on the sphere and applying the same process to a square in the plane (with an initial 8 half-square triangles representing octants) and then mapping each resulting spherical triangle to the corresponding planar triangle. The way the hierarchical triangular mesh works is to start with a triangle, and then bisect each side, connecting the new points with straight segments (great circle arcs on the sphere) which splits the original triangle into 4 sub-triangles. This process is then iterated.

Unfortunately this is actually a fairly crummy method if you want to bisect very many times, causing a lot of weird distortion. As you found out, it does not have much of anything to do with the Pierce projection or any other smooth projection, because it's based on a discrete process.

There are a number of other map projections based on starting with octants; I don't think a comparison really belongs at this page – it seems at least moderately off topic – but it might be nice to make a new article somewhere with such a comparison. –jacobolus (t) 19:24, 4 October 2024 (UTC)