HELP! How to turn a circle into a square?

[Caenwyr]

Okay, the title was a bit dramatic maybe, but it got you clicking, didn't it . Basically I'm out of inspiration for this terrible conundrum.

What I'm after is the following. I want a way to stretch a circle - horizontally only - so that it fits within a square. Or very practically, I want to turn this:

(the Apian-II double hemisphere projection)

into this:

(the equirectangular projection)

I know it can easily be done the other way around, e.g. using NASA's awesome freeware tool G.Projector (which is what I used to turn the second image into the first one!), but I can't think of a way to do it in the direction I want: from Apian-II to equirectangular.

Basically what I'd need is a way to take each disc in that first image, and stretch each of the pixel rows horizontally with such an amount that they all end up just as wide as the equator. Obviously the stretch factor would depend on how far away that particular pixel row is from the "equator row". Then I just need to stitch both squares et voilà, I'm back at the equirectangular!

For each row of pixels, the stretch factor would be N_equator / N_actual, with

• N_equator being the number of pixels on the equator, which is known, and
• N_actual being the sine of the angle that has that particular row height (measured from the equator, as a fraction of the height of the poles) as its cosine.

That last bit sounds complicated but actually you just take the sine of acos (H_actual / H_pole), with

• H_actual being the vertical distance between the equator row and the actual row, and
• H_pole being the vertical distance between the equator and the pole, which is known (and identical to N_equator since we're talking about a perfect circle here!)

But "simple" as that may sound, I have no knowledge of coding, and no idea whatsoever of how I can turn this into something that actually does the thing I want, no matter how long I ponder on it...

So all in all, I'm at a loss. It's not a terrible inconvenience, I should be able to work around it, but man would it be awesome to turn my Apian-II map into an equirectangular one again...

[Azélor]

I'm not sure what you are trying to reproject here, but it will stretch your map so bad that you will have to redraw it.

[Caenwyr]

I'm not sure what you are trying to reproject here, but it will stretch your map so bad that you will have to redraw it.

Well no, it should simply turn that first map (Apian-II) into that second map (Equirectangular). No horrible stretching anywhere, just a different projection.

Anyone else has an idea?

[Azélor]

It can work if your map is georeferenced which is not a problem if you are working with the Earth.

I think this thread could be helpful for you : https://www.cartographersguild.com/s...049#post395049

[waldronate]

Apian II can't be transformed into Equirectangular by simply stretching things horizontally, if I'm reading it correctly. Depending on your accuracy needs, you might be able to use Mollweide instead.

My elderly Windows program ReprojectImage ( http://www.fracterra.com/ReprojectImage.zip ) can do something like this:


If you look closely, you'll see that there is a bit of stretch in the polar regions. Doing this process twice and assembling the pieces in something like Photoshop should get you most of the way there if you can't find an exact transform.

[Caenwyr]

Apian II can't be transformed into Equirectangular by simply stretching things horizontally, if I'm reading it correctly. Depending on your accuracy needs, you might be able to use Mollweide instead.

My elderly Windows program ReprojectImage ( http://www.fracterra.com/ReprojectImage.zip ) can do something like this:


If you look closely, you'll see that there is a bit of stretch in the polar regions. Doing this process twice and assembling the pieces in something like Photoshop should get you most of the way there if you can't find an exact transform.

Sorry for not answering earlier, Waldronate, but man, you saved me here! Thanks a bunch!!

[waldronate]

Glad that I was able to help. It's nice when one of my elderly pieces of software can get out and stretch its legs a little.