  The uniqueness of the orthogonal Z^\gamma-circle patterns as studied by
Bobenko and Agafonov is shown, given the combinatorics and some boundary
conditions. Furthermore we study (infinite) rhombic embeddings in the plane
which are quasicrystallic, that is they have only finitely many different edge
directions. Bicoloring the vertices of the rhombi and adding circles with
centers at vertices of one of the colors and radius equal to the edge length
leads to isoradial quasicrystallic circle patterns. We prove for a large class
of such circle patterns which cover the whole plane that they are uniquely
determined up to affine transformations by the combinatorics and the
intersection angles. Combining these two results, we obtain the rigidity of
large classes of quasicrystallic Z^\gamma-circle patterns.
