We study one variable meromorphic functions mapping a planar real algebraic setAAto another real algebraic set in the complex plane. By using the theory of Schwarz reflection functions, we show that for certainAA, these meromorphic functions must be rational. In particular, whenAAis 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 for2m−12m-1dimensional sphere inCm\mathbb {C}^mwhenm≥2m\ge 2.