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.
Okay so I was hoping that I could use a rational polynomial that's asymptomatically linear, by having the denominator one degree less than the numerator. Which would work fine over the domain of the real numbers. But because a polynomial of degree n always has n zeroes in the complex plane, then a rational polynomial will always introduce at least one point in the complex domain where the output blows up to infinity. And that's worse than the polynomial growth I was trying to avoid in the first place.
Are there no continuous, analytical functions that are one-to-one in the complex domain besides simple degree-one polynomials?