An Explicit Forward–Inverse Dictionary for Quadratic Miquel Motions

We establish an explicit forward–inverse dictionary for the genuinely quadratic, nondegenerate branch of Miquel motions, rather than a classification of general rational quartics. For three side points moving with prescribed polynomial ratios on a fixed nondegenerate triangle, the reduced Miquel map in this branch has, after normalization, a linear numerator and quadratic denominator. Its locus is an irreducible real bicircular quartic. In the forward direction, one parameter set determines the canonical motion, the unique finite singularity, two singular double-contact lines and the real singularity type. The discriminant of a quadratic preimage polynomial separates real nodes, ordinary cusps and isolated real nodes; a point-geometric perpendicularity condition is equivalent to orthogonality of the two double-contact lines. In the inverse direction, the singular double-contact data determine the canonical parameters and explicit polynomial side-point ratios on a real triangle. Exchanging the two double-contact directions gives a second realization related to the first by the reciprocal parameter map. We separately catalog reducible, common-factor and lower-degree degenerations. Exact symbolic checks and reproducible examples accompany all principal formulas. The contribution is a constructive two-way correspondence between one-dimensional quadratic Miquel motions and classical invariants of their real algebraic loci.