Tangents-free property in invariant disk packings generated by some general planar piecewise isometries

In this paper we investigate properties of invariant disk packings in the dynamics of some general planar piecewise isometries. By introducing codings underlying the map operations, we can give explicit expressions for the centers of the disks by analytic functions of the parameters, and then show that tangents free in invariant disk packing is a common property of the dynamics of piecewise isometries arising from various backgrounds. This supports the long-standing conjecture that the complements of the packings have positive Lebesgue measure.