Fermat's spiral and the line between Yin and Yang

Let $D$ denote a disk of unit area. We call a subset $A$ of $D$ perfect if it has measure 1/2 and, with respect to any axial symmetry of $D$, the maximal symmetric subset of $A$ has measure 1/4. We call a curve $β$ in $D$ an yin-yang line if $β$ splits $D$ into two congruent perfect sets, $β$ crosses each concentric circle of $D$ twice, $β$ crosses each radius of $D$ once. We prove that Fermat's spiral is a unique yin-yang line in the class of smooth curves algebraic in polar coordinates.

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