  We consider Apollonian circle packings of a half Euclidean plane. We give
necessary and sufficient conditions for two such packings to be related by a
Euclidean similarity (that is, by translations, reflections, rotations and
dilations) and describe explicitly the group of self-similarities of a given
packing. We observe that packings with a non-trivial self-similarity correspond
to positive real numbers that are the roots of quadratic polynomials with
rational coefficients. This is reflected in a close connection between
Apollonian circle packings and continued fractions which allows us to
completely classify such packings up to similarity.
