Constructing quasiconformal maps using circle packings and Brooks's parameterization of quadrilaterals

Circle packings have deep and well-established connections to conformal maps. Some methods for using circle packings to approximate quasiconformal maps have been studied, but they are not directly tied to the circle geometry. We present here a means to construct quasiconformal maps using Brooks’s parameterization of quadrilateral regions bounded by circles. The Brooks parameter acts as a sort of circle packing module, allowing us to directly affect the complex dilatation of our quasiconformal maps.