Recently J. Mateu, J. Orobitg, and J. Verdera showed that a Hölder continuous complex dilatation supported on smooth domains is a sufficient condition for the resulting quasiconformal map to be bi-Lipschitz. Their proof is analytic and based on properties of the Beurling-Ahlfors transform. We give an alternate, more geometric proof and use it to extend their result to supporting domains with positive angle corners.