Hammond optimal conformal projection

[daan]

I suppose the reason is that the Hotine is already built into Geocart... otherwise, I would have asked, since you are distorting it anyways via a sixth-order polynomial, why you bothered to use the Transverse Mercator for the ellipsoid rather than just the simple spherical one.

Hotine is defined for oblique parameterization, which is the only reason I used it: It already has built into its parameterization exactly what I wanted. In point of fact, I use the spherical development due to Miller’s definition against the sphere. Therefore, it is no different than using an oblique Mercator via spherical transformations. Should I switch to ellipsoidal, I already have everything in place to do so.

(Of course, if areal distortions are to be minimized, the actual areal distortions, rather than imagined ones, should be minimized, and the resulting projection should really be conformal - although that last could be assured merely by starting with the Rosenmund.)

Same logic applies to optimizing the match to the bipolar oblique conformal’s distortion vs minimizing the distortion against the ellipsoid. In this proof-of-concept, I merely messed around to approximate the distortion pattern of Miller’s invention, but if I were to write the code to optimize something, I wouldn’t waste it on replicating Miller’s bipolar conic. I’d instead optimize the regions of interest.

— daan

[daan]

8th order polynomial getting closer to bipolar’s distortion pattern, with the completely chance benefit of rehabilitating Hawaii:

8th order, good Hawaii

bipolar8.1.jpg (166.26 KiB) Viewed 1355 times

Throwing Hawaii under the bus, this 8th order gives as good or better than Miller’s pretty much everywhere:

8th order, better than Miller

bipolar8.2.jpg (163.54 KiB) Viewed 1355 times

But the latter has not been optimized to lower distortion throughout the Americas. It could do better for Labrador/Greenland.
— daan

[quadibloc]

Amazing! If I understand you correctly, though, you did not use the same basic technique in obtaining the coefficients for these polynomials as was used for optimized projections like the GS50 projection. What is the technique you are using?

[daan]

In the general case, I am just telling the polynomial to map a set of (somewhat knowledgeably chosen) n+1 points from Hotine to the same coordinates as found on the Miller bipolar. This is to reconstruct the Miller bipolar using polynomial of order n. Since I have a fair amount of experience with this stuff, I have a sense of what points might be good to choose for the set. Plenty of “just trying things” involved.

To improve the distortion pattern beyond Miller’s bipolar, I can’t rely on Miller’s mapped points, of course. I only moved one point away from Miller’s coordinates to get the second case, which was the southern tip of Greenland. How did I know where to move it to? I looked at the distortion pattern at that location to estimate the distance on the ground that the Miller distortion moved it away from optimal, adjusted it back in the opposite direction that the distortion pattern indicated it had moved it, iterated with manual adjustments a few times, and satisfied myself that more trials on that one point were not going to improve things meaningfully. A global optimization would be needed. I would not undertake that on the spherical model, where this is all just fake anyway. Maybe I will look at a real implementation sometime. The optimization iteration is not particularly complicated; I just haven’t bought into the project, particularly. Just been messing around while working on stuff more meaningful to me.

Early on I thought I could get close to Miller bipolar using an oscillating function (sine or something), but it was just turning into a quagmire.

— daan