Converting icosahedral hex-map to Plate-Carree (equirectangular)

[MollyJ]

I have been trying to find map projection software that can convert icosahedral to equirectangular for making pov-ray planet images. I've had no luck at all so far. It may be that I am using the wrong search terms or that this feature does exist in some suite as a drop-down menu so obscure that no one knows about it. Does this sound familiar to anyone here? Is this something that you would want to explore as an idea? I attached image of the input and output maps for clarity. Any good coders here interested in the project? I already have all the maths needed. - Molly_J
icosa-2-PlateCarree.jpg

[MollyJ]

I forgot to attach the globe image generated from the flat map on the lower left.
hexcarree-globe.jpg

[waldronate]

It depends in large part on the precise nature of your icosahedral projection. On your first image, you show two rather different icosahedral views: the "Traveller" hexagonal one that is designed to get the idea across, but isn't really a nicely defined projection as such; and a nice icosahedral one that looks to be twenty piecewise stereographic projections arranged into a generally similar shape. The vertical split on that last face on the Traveller map is designed to make things fit more nicely on a rectangular page, but it's a little annoying to calculate. Fractal Terrains ( http://www.profantasy.com ) can export the Traveller-style thing using its Cosmographer template feature ( File>>Export World>>Cosmographer Template ), but it can't go the other way.
Wilbur ( http://fracterra.com/software.html ) has had a piecewise linear transform that converts an Equirectangular map into something that folds into an icosahedron for a couple of decades now using the Texture>>Other Maps>>Icosahedral Projection menu option. It also goes the other way if you hold down the Shift key when using the Texture>>Other Maps>>Icosahedral Projection menu option. To a good approximation, it's an interrupted Collignon projection.
It looks like your conversion is using a piecewise Stereographic (or some related projection with curved parallels) projection to convert the Traveller map to Equirectangular. This kind of conversion generates unexpected results if you (like too many people I've asked over the years) assume that those straight lines between segments are lines of latitude, which would make the whole thing some kind of projection in the cylindrical family. The conversion that you're showing here converts those straight lines along the Traveller map segments into curves on the flat map rather than keeping them straight, which will only matter for specific uses of the map. The hexagons are also distorted on the converted flat map, which may be the exact right thing or it might be bad, depending on the player in question.
I guess it all comes down to deciding what the Traveller map means. I've seen a folks get out a ruler and measure things on the hex maps, for the couple of times that they wanted a distance. It's a tough thing to answer, because the maps are generally so coarse and notional that the distortions map no real difference in accuracy for linear or non-linear.
And yes, it does look like I've completed my transition to crusty old curmudgeon.

[su_liam]

The longer, more detailed instructions are somewhere in my blog. I’m still at work, so I can’t find it right now, but I’m heading home soon…

[su_liam]

I didn’t know you could do that inverse function in Wilbur. Wild. Not knowing that, I put way too much effort into figuring out the problem for myself. That hand-drawn looking globe in the upper right was the product of that work. I’m very familiar with Korslan and Marditha and somewhere in the darkness beyond the Kors Gulf, Hardlannen.
Short story. I made an ico-sphere in Blender(reduced to 20 sides). I fit a UV-map to my icosahedral image, used that to map my image as a texture on the icosphere. Finally, I put a panorama camera set to equirectangular and a light source at the center of the icosphere and rendered.

[su_liam]

The blog I currently have, is roundabout and uses 3 free apps, ‘cause I didn’t fully understand what I was doing. It looks like it’s time to write a new tutorial. In the meantime, here’s what I got.