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.