Hammond optimal conformal projection

[Atarimaster]

Hello,

in Flattening the Earth, Synder mentions the Hammond optimal conformal projection (Page 246) which is suited for continental maps.
It sounds interesting, but I couldn’t find any further information about it on the web.

Have the formulae even been disclosed or are they kept under wraps, or maybe patented?

Regards,
Tobias

[daan]

Ooh, nice topic!

I actually have a copy of Hammond’s atlas using the projection. Hammond contracted with Mitch Feigenbaum (of chaos theory fame) to do the mathematical work for the projection. The projection is based on Chebyshev’s criteria for the optimal conformal map. Specifically, the definition of optimality is the map with the minimum ratio of greatest to least scale factor within the region of interest (min/max criterion). Chebyshev conjectured (1853), and was proved by D.A. Gave (sometimes written Grawe) (1896), that the min/max criterion is met when the region of interest is bounded by a path of constant scale.

John P. Snyder himself worked on this problem in the 1980s but was only able to arrive at approximations to optimal (see his GS50 projection). Snyder met with Feigenbaum and the Hammond people in the mid 90s to talk about the development, and came away awed, but as far as I know, nothing was ever published by Hammond or Feigenbaum other than the atlas itself. The atlas describes the optimality of the maps but says little about the method. Ivan Nestorov published on the technique in the late 90s, and many papers have been written using the technique in specific ways over the years since. It’s wrapped up in solving the Dirichtlet boundary condition. In general, the solution wouldn’t be a “formula”, but, rather, a technique.

— daan

[Atarimaster]

Thank you, daan, for providing the information – that’s very interesting.
Of course, it’d be great to have the projection in Geocart some day, but from the way it sounds, I better don't get my hopes up.

Regards,
Tobias

[daan]

Of course, it’d be great to have the projection in Geocart some day, but from the way it sounds, I better don't get my hopes up.

Since it is a technique, rather than a "projection", it would be very hard to add in any general way. Just defining the boundary you want for optimality is already a user interface nightmare. From there, the computational process to optimize the projection for that boundary is huge. For specifically delineated regions, much of the computation could happen in advance, with results saved away per projection. For just any arbitrary region, however, I think your pessimism is justified.

— daan

[quadibloc]

I just today heard about these projections, by reading the paper by Goldberg and Gott on large-scale distortions.

This sounds exciting and interesting to me for one specific reason: the fact that the Bipolar Oblique Conic Conformal projection has to be stitched together in the middle with a small region that isn't conformal has always been a disappointment, and I felt that with all the techniques that exist to produce conformal projections, something fully conformal, but closer to it than an oblique Mercator, ought to be possible, and is something I would like to see.

But I would be happy to settle for an approximate optimization, like GS50, rather than seeking perfect optimization like the Eisenlohr.

[daan]

The fact that the Bipolar Oblique Conic Conformal projection has to be stitched together in the middle with a small region that isn't conformal has always been a disappointment, and I felt that with all the techniques that exist to produce conformal projections, something fully conformal, but closer to it than an oblique Mercator, ought to be possible, and is something I would like to see.

This should be feasible. I would start with an oblique Mercator, find some simple (complex, of course) function to bend the the axis to something roughly coinciding with the bipolar oblique’s two central meridians along most of its length, and then use a large-degree polynomial (or probably FFT) to change the parallel spacing along the axis to something approximating the bipolar oblique’s—or even better than, if you were to invest in some analysis to approximate optimal better than the bipolar oblique.

— daan

[quadibloc]

For now, I haven't attempted yet to pursue making a conformal projection of this type.

However, I have done one related, but much simpler, thing. I've made some additions to my page on the Polyconic projection. I've illustrated the result when it is misused to draw a map of the entire (well, almost; the lower 48) United States, and also illustrated an ambitious use of the Polyconic for a National Geographic map of Asia in 1942, having seen it mentioned in "The Round Earth on Flat Paper".

So I tried my hand at applying an Oblique Polyconic where the Oblique Mercator had already succeeded, to a map of the Americas, and got the following:

[daan]

That oblique American polyconic is quite good. I think the only way to improve it would be along the lines noted above: keep the axis of low distortion in the center of the regions of interest by having it bend rather sharply in the Caribbean. That’s more a job for a conformal map, though. I can’t think of how to do such a thing in an orderly way with a conventional projection, off the top of my head.

— daan

[daan]

The fact that the Bipolar Oblique Conic Conformal projection has to be stitched together in the middle with a small region that isn't conformal has always been a disappointment, and I felt that with all the techniques that exist to produce conformal projections, something fully conformal, but closer to it than an oblique Mercator, ought to be possible, and is something I would like to see.

This should be feasible.

Decided to try this. Nothing fancy: Hotine distorted via a 6th order polynomial; no attempt to optimize via numerical means. Hence, it’s just a proof-of-concept and very likely could be considerably refined.

Modified Hotine vs bipolar oblique conic conformal

bipolar.jpg (327.19 KiB) Viewed 1453 times

— daan

[quadibloc]

Decided to try this. Nothing fancy: Hotine distorted via a 6th order polynomial; no attempt to optimize via numerical means. Hence, it’s just a proof-of-concept and very likely could be considerably refined.

None the less, I'm certainly impressed!

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. (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.) Of course, now I'm going to have to read up and find out enough about the Hotine, Laborde, and Rosenmund oblique Mercators so that I can intelligently describe the distinction between them... as well as versions of the Transverse Mercator for the ellipsoid other than the Gauss-Kruger!

Snyder, it turns out, explained them well enough in Flattening the Earth, so that I could understand them as much as I needed for my purposes. Rosenmund's 1903 version simply used the conformal latitude, and the 1928 Laborde version was too complicated for me to discuss in any detail.