We study one variable meromorphic functions mapping a planar real algebraic set
A
A
to another real algebraic set in the complex plane. By using the theory of Schwarz reflection functions, we show that for certain
A
A
, these meromorphic functions must be rational. In particular, when
A
A
is the standard unit circle, we obtain a one dimensional analog of Poincaré [Acta Math. 2 (1883), pp. 97–113], Tanaka [J. Math. Soc. Japan 14 (1962), pp. 397–429] and Alexander’s [Math. Ann. 209 (1974), pp. 249–256] rationality results for
2
m
−
1
2m-1
dimensional sphere in
C
m
\mathbb {C}^m
when
m
≥
2
m\ge 2
.