HallsofIvy: Thanks, that's a nice bijection. I clearly need to think more geometrically for this type of problem.
mathwonk: I don't think there is a problem at x=0 (for Halls' map) because you can just define r to be 1 for x=0 and then it is smooth on S1.
Hi, I have been told that in R^2 the unit circle {(x,y) | x^2 + y^2 = 1} is smoothly mappable to the curve {(x,y) | x^4 + y^2 = 1}.
Can someone please tell me what this smooth map is between them? I can only think of using the map (x,y) --> (sqrt(x), y) if x is non-negative and (sqrt(-x), y)...