Conformal polyhedral maps

[daan]

Talking about polyhedral maps – would it be tough to integrate some equal-area polyhedral maps using the van Leeuwen projection?
For example, maybe a dymaxion-like equivalent map, or configurations similar to Wijk’s “optimal fold-outs”?

Compared to the grueling, massive amount of work I put into the conformal projections? NO.


But, still, a fair amount of work.

— daan

[Atarimaster]

But, still, a fair amount of work.

I had a hunch you’d say that. That’s why I asked for the amount of work instead of just adding them in the Wish Lists board.
Well, it would be nice to compare them to the conformals, but they’re not on top of my "most wanted" list so I guess it’s not worth the effort as far as I’m concerned

Kind regards,
Tobias

[Maezar]

Just checking to see if this feature is something I can download yet. Can I beta test it maybe?

[quadibloc]

Compared to the grueling, massive amount of work I put into the conformal projections?

Actually, there you are. It is true that conformal polyhedral projections are better than Gnomonic polyhedral projections. But the math involved for the Gnomonic ones is so much simpler that it's hardly surprising that the conformal ones have only been attempted in relatively recent years.

[daan]

Just checking to see if this feature is something I can download yet. Can I beta test it maybe?

Oh, I hadn’t comitted to working on it. I’ll add it to the list of requests. Maybe don’t hold your breath!

— daan

[quadibloc]

Unfortunately, conformal maps are not possible for two different shapes if they must match along an edge. The only exceptions are:
• The shapes are rotations and/or reflections of each other; or
• One shape is just a piece of the other shape.

You are quite right for the case where what is mapped to each shape is decided in advance. (Actually, I suspect it's more like "really difficult" and not impossible, since we have both August's conformal and the Eisenlohr.) But if you join the two shapes together to form a single shape before you do the mapping, then make your cut afterwards...

Ah, but you're probably talking about a more heavily constrained case than that. Where the two different shapes are matching along all edges - so as to, say, provide a conformal polyhedral map on an Archimedian solid, such as the cuboctahedron - to match Fuller's original Dymaxion map. (Again, if you can do for each polygon what was done in the Eisenlohr for the two-cusped epicycloid, however, you could do it, but I think the math is pretty hairy.)

Ah, wait a moment - given that the Eisenlohr has a bigger disparity of minimum and maximum scale than August's conformal, squares and triangles for a cuboctahedral projection would likely have the same defect - and that would make the result unattractive as a polyhedral projection. So not only would it be really difficult, but it's also not worth doing.

[daan]

Unfortunately, conformal maps are not possible for two different shapes if they must match along an edge. The only exceptions are:
• The shapes are rotations and/or reflections of each other; or
• One shape is just a piece of the other shape.

You are quite right for the case where what is mapped to each shape is decided in advance… But if you join the two shapes together to form a single shape before you do the mapping, then make your cut afterwards...

I would consider that to be the second case, where one shape is just a piece of the other shape (or the union of the shapes). But yes, you can map to pretty much any shape and then cut it up to get pieces that you like.

…(Actually, I suspect it's more like "really difficult" and not impossible, since we have both August's conformal and the Eisenlohr.)… (Again, if you can do for each polygon what was done in the Eisenlohr for the two-cusped epicycloid, however, you could do it, but I think the math is pretty hairy.)

Please elaborate. I don’t see how Eisenlohr/August fit into this. They do not match along any mapped edges.

— daan

[quadibloc]

Please elaborate. I don’t see how Eisenlohr/August fit into this. They do not match along any mapped edges.

No, but they have the same shape. And the Eisenlohr has the property that the scale is uniform over the whole edge.

I suspect that it isn't just for the two-cusped epicycloid that this can be done, but any shape. And if the perimeters are appropriately scaled, doing that for the different types of face for a polyhedral map based on an Archimedian solid would be possible. But I suspect the results wouldn't be worthwhile, as the scale would vary to a greater extent from the center to the edges of the face in order to meet that constraint than for the more prosaic conformal mappings used with ordinary polynedral maps.

I just realized that I am mistaken. How the Eisenlohr was managed, I don't know, but when one is dealing with polygons with straight-line edges, the only way you can have uniform scale is for the identity function to the plane, and Mercator to the sphere, thanks to analytic continuation. So you are right, the control over edge spacing is very limited.

[daan]

Please elaborate. I don’t see how Eisenlohr/August fit into this. They do not match along any mapped edges.

No, but they have the same shape.

They… don’t. (However, even having the same shape would not imply that they must be the same map by the strict rules I have been talking about; see (3) below.)

if the perimeters are appropriately scaled, doing that for the different types of face for a polyhedral map based on an Archimedian solid would be possible.

Let’s get more specific. Everything I write here presupposes conformal mapping only.

1. Fold the Archimedean solid out flat. Map the entire globe conformally onto the entire fold-out. This yields edges that match.

2. Or, pick a face A of the solid, and give yourself the luxury of deciding which region of the sphere will map to it—such as, A represents a 3/64 portion of the globe. Unless you choose this to match what (1) gives you exactly, you will not be able to map the solid conformally. There are no mappings for the other faces that will match along all edges.

3. Being able to choose the outer shape is not the same as determining the entire conformal map from an edge. It is possible to map the entire sphere onto a square in different ways (for example), beyond just scaling or rotation of the map or spherical rotations of the preimage. This can happen because the spacing of preimage points along the edges will differ between the maps, as well as which preimage points the edge consists of. See, for example, Cox’s projection of the world into a triangle, versus the conformal tetrahedron of Lee. This still means that the smallest finite segment of the map determines the entire map: Not just its shape, but how points are distributed along the shape’s edges as well as the entire interior. The exception here is any branch cuts; those are matters of choice, and of course will affect the edge.

— daan

[quadibloc]

Fold the Archimedean solid out flat. Map the entire globe conformally onto the entire fold-out. This yields edges that match.

Yes. And, because Cox's conformal projection of the world on a triangle does not qualify as a truly polyhedral conformal projection of the world on a tetrahedron... I have a symmetry argument for you.

A conformal mapping of the sphere onto a cuboctahedron where all the squares are the same, and all the triangles are the same? Yes, that's possible. How you do that: divide the squares into four triangles, and the triangles into three triangles... to make 24 kites. Divide the sphere into 24 identical regions, map them conformally to the kites, then rebuild the squares and the triangles from those kites.

This won't apply to all Archimedian solids, because they are only guaranteed to have identical vertices, not edges. But it may be possible to do something similar by cutting pieces of the polygons that meet at a vertex.