All right then.Atarimaster wrote:Will you at some point (here or elsewhere) elaborate on your ideas on the optimal equal-area map?
<speculation>
Snyder was on the right track, but I don’t think he grasped that the boundary condition means nothing to the interior of an equal-area projection. The reason for this difference between conformal and equal-area properties is that equal-area projections are less constrained than conformal. Let’s see how.
For a plane-to-plane projection to be conformal, it must meet two conditions:
∂x′/∂x = ∂y′/∂y
∂x′/∂y = – ∂y′/∂x
Equal-area projections, on the other hand, only need to meet one condition:
(∂x′/∂x)(∂y′/∂y) – (∂y′/∂x)(∂x′/∂y) = 1
The Wiechel example I showed above was not even as general than it could have been. We could take an arbitrary little circle somewhere in the Lambert azimuthal and twist up its contents while leaving the entire rest of the map alone, and we’d still have an equal-area projection. That’s completely impossible with a conformal projection: Anything we do to the tiniest part of a conformal projection affects the entire map.
We cannot have a theory of optimality without a theory of uniqueness for the mappings in the space we are working in; otherwise we would not be able to say, “This map is optimal”. A theory of uniqueness means we need something more to constrain equal-area maps; they’re too “floppy”, otherwise.
Let’s posit that, amongst other things, an optimal equal-area map does need to be bounded by an isocol. Let’s posit that because (a) it seems reasonable; and (b) empirical evidence suggests the same thing, given Snyder’s results, and others’. Referring back to the Wiechel vs. Lambert azimuthal example, what is it, really, about the Wiechel that makes it so much worse?
You can answer that it’s the twist, and that wouldn’t be wrong, but how do you even measure “twist”? What is twisted against what? It’s obvious in the polar view that the meridians twist—but what is it about that that makes it “bad”? Meridians in an optimal projection will certainly curve, so it’s not about the curvature per se. Any projection’s meridians curve if you change the aspect, for that matter. Furthermore, if we switch Wiechel to some arbitrary aspect, I bet a lot of people wouldn’t even realize that it’s “twisted”:
- Wiechel oblique
- Wiechel map.jpg (165.17 KiB) Viewed 90906 times
Or maybe there’s something else going on. I think you can measure a thing called “torsion” that is not directly related to angular deformation. I define it like this: Torsion is the departure of the major axis of the Tissot ellipse from the isocol. If we superpose the Tissot ellipses onto both the Wiechel and the Lambert azimuthal, you can see that the major axis of the ellipses on the Lambert lie on the isocols, whereas on the Wiechel, they are skewed away from it.
- Wiechel on left; Lambert on right
- Wiechel+Lambert.jpg (259.19 KiB) Viewed 89925 times
What this implies is that there is an equal-area analog to any conformal map such that the equal-area map has isocols coinciding with the isocols of the conformal map that is optimal for the same region. The set of equal-area maps that could ever be optimal for a region is one cardinality below the complete class of equal-area projections, much like integers are one cardinality below real numbers. Pseudocylindric projections cannot be optimal, nor even close. Cylindric and conic projections cannot be optimal because they do not bound regions with isocols.
The reason I think this works, and works as a theory, is that it constrains the equal-area maps in clear ways that prohibit deviant behavior such as localized twists. It also avoids controversy or alternatives over how to measure an aggregate or quantity across a region.
</speculation>
Unfortunately, I have made no progress on a proof; nor have I made any progress on feasible numerical methods for generating maps based on this conjecture. I have done a huge number of experiments that ended up disproving ancillary conjectures, and unfortunately, those disproofs eliminated various ideas I had for tractable computational methods.
There are other implications to my conjecture that I won’t get into, particularly when it comes to “lumpy” maps where there are multiple, disorganized regions of low distortion interspersed with high distortion, such as that GS50 projection I show in an earlier post in this thread.
— daan
Nice catch.