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We obtain a first order differential equation for the driving function of thechordal Loewner differential equation in the case where the domain is slit by acurve which is a trajectory arc of certain quadratic differentials. Inparticular this includes the case when the curve is a path on the square,triangle or hexagonal lattice in the upper halfplane or, indeed, in any domainwith boundary on the lattice. We also demonstrate how we use this to calculatethe driving function numerically. Equivalent results for other variants of theLoewner differential equation are also obtained: Multiple slits in the chordalLoewner differential equation and the radial Loewner differential equation. Themethod also works for other versions of the Loewner differential equation. Theproof of our formula uses a generalization of Schwarz-Christoffel mapping todomains bounded by trajectory arcs of rotations of a given quadraticdifferential that is of interest in its own right.

The Loewner driving function of trajectory arcs of quadratic differentials