I want to design a family of conformal map projections where the perimeter of the rectangular map has nearly constant scale (thus optimizing the overall scale distortion in the interior of the rectangle) at a range of extents and aspect ratios. I think this can probably be accomplished with a 5th-order or 7th-order (or maybe even 9th-order) complex maybe-rational polynomial transformation of a stereographic azimuthal map. The tricky part will be developing a general procedure to optimize the the polynomial for a given rectangle size. The hard part will be finding a general formula for the optimal polynomial (or a good approximation thereof) in terms of the desired map extents.
In another rechost-addendum, I determined that a rational polynomial will not fit my needs. Furthermore, I can derive some constraints on the polynomial coefficients rather easily based on some simple desired properties:
The polynomial maps real numbers to real numbers
All polynomial coefficients are real.
The polynomial maps pure-imaginary numbers to pure-imaginary numbers
All even-degree polynomial coefficients are zero.
The neighborhood about the origin is not scaled or rotated by the polynomial transformation
The first-degree coefficient is 1.
Swapping the extent parameters (a, b) has the effect of applying the transformation in a 90°-rotated view
f(b, a, z) = i·f(a, b, −i·z)
The third-degree coefficient, seventh-degree coefficient, and so on, are negated if a and b are swapped, and zero if a equals b. The fifth-degree coefficient, ninth-degree coefficient, and so on, are unchanged if a and b are swapped.
My projection family has a name! Also, I've now got a simple Python program to try different polynomials with, and plot a simple graticule (centered on the equator, because azimuthal map projections in polar aspects are kinda boring) in the transformed projection with lines of constant scale. Just messing around with manually entering polynomials (and their derivatives), I found that f(z) = 0.04z5 − 0.05z3 + z makes a pretty nearly-rectangular curve of constant scale (which happens to be a little over 5% bigger than at the origin) which would be an appropriate result for extent parameters a = 0.63 and b = 0.35, or about 5000 × 2800 miles on Earth; this is shown in the first attached image. Then I plotted f(z) = 0.04z5 + 0.05z3 + z for the transverse orientation of the same resulting projection, shown in the second attached image. In both images, the latitude/longitude graticule shows increments of 15°, and the x and y axes are scaled so that one unit of projected distance nominally represents one radian of central angle, or one globe-radius of surface distance (3963 miles on Earth).