Actually, the spiral curve as it's used in transportation engineering has a pretty well-defined model - no need to solve for it generally... Unless it's a more elegant solution? Anyway, a brief overview of the parameters and their mathematical definitions for a transportation engineering clothoid can be found here:

https://www.mathalino.com/reviewer/surv ... iral-curve

Given those parameters, I just need to sort out which ones need to be given by the user, and which can be calculated.

To this point, I know the distance between PI's, the simple curve radius, and the central angle of the simple curve. From that, I can probably get the spiral's tangent length (Ts), but to get the other parameters, I'll probably need to know one or more of the following: the spiral angle, the circular curve offset (length of throw), and the length of the spiral curve... That's just a guess, though.

If you can think of a better approach, though, I'd love to hear it.