Boundary parametrization of planar self-affine tiles with collinear digit set

We consider a class of planar self-affine tiles T = M-1 boolean OR(a is an element of D)(T + a) generated by an expanding integral matrix M and a collinear digit set D as follows: M = (0 -B 1 -A), D = {(0 0), ... , (vertical bar B vertical bar - 1 0)}. We give a parametrization S-1 -> partial derivative T of the boundary of T with the following standard properties. It is Holder continuous and associated with a sequence of simple closed polygonal approximations whose vertices lie on partial derivative T and have algebraic preimages. We derive a new proof that T is homeomorphic to a disk if and only if 2 vertical bar A vertical bar <=vertical bar B + 2 vertical bar.