Algorithmic search for inverse projections about d3-geo-projection

[Fil]

d3-geo-projection now has newton.js which can help in some cases

[Fil]

https://marcinciura.wordpress.com/2016/11/20/authagraph/ has this note:

As inverting the equations of the projection algebraically is a hopeless task, I resorted to the numerical Broyden’s method. The initial guess of (\phi, \lambda) = f^{-1}(x, y) comes from inverting the gnomonic projection. When the accuracy is set to 10-7 R, i.e. about one metre, the method usually finds \phi and \lambda after 5 iterations.

So it looks like a nice generic method for this would be to use a known inverse proj as a first step, then optimize on the real stuff: unknown.invert = approximation_method (known.invert);

[Fil]

@mciura I cited you here also

[MarcinCiura]

In my experience, Broyden's method had the median number of iterations equal to 5, and the long tail ended at 18 or 19 iterations. I had to come up with nontrivial tricks to make it work near the poles — I suppose Newton–Raphson and any other method would need them too.

[Fil]

Here's a function that apparently works correctly for the inverse of:

• ChamberlinAfrica
• ModifiedStereographicAlaska
• ModifiedStereographicGs48
• ModifiedStereographicMiller
• ModifiedStereographicLee
• TwoPointAzimuthalUsa
• ModifiedStereographicGs50 (the part over Europe is broken but it's not supposed to be shown)
• InterruptedSinuMollweide (needs a bit more syntactic sugar to go around interrupt.invert, see block)
• Bertin 1953 (work in progress)

https://bl.ocks.org/Fil/07930ee04d7fbd9e5e3e84e803d35dcf

[Fil]

This inverse function now works on all remaining d3 projections.

The code would need to be cleaned up before merging, and before I try and do that, I'd like to know if you think the 2D (and spherical) Newton-Raphson method should be promoted into its own plugin (exporting d3.solve & d3.geoSolve), rather than be constrained to d3-geo.

There's also a problem of convergence, evidenced by the red dots on https://bl.ocks.org/Fil/07930ee04d7fbd9e5e3e84e803d35dcf

This might be fixed by restricting the search domain to the initial clipAngle.