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We study one variable meromorphic functions mapping a planar real algebraic set A to another real algebraic set in the complex plane. By using the theory of Schwarz reflection functions, we show that for certain A, these meromorphic functions must be rational. In particular, when 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 2m-1 dimensional sphere in \mathbb {C}^m when m\ge 2. [ABSTRACT FROM AUTHOR]