  In this paper, we consider the self-affinity of planar curves. It is regarded
as an important property to characterize the log-aesthetic curves which have
been studied as reference curves or guidelines for designing aesthetic shapes
in CAD systems. We reformulate the two different self-affinities proposed in
the development of log-aesthetic curves. We give rigorous proof that one
self-affinity actually characterizes log-aesthetic curves, while another one
characterizes parabolas. We then propose a new self-affinity which, in
equiaffine geometry, characterizes the constant curvature curves (the quadratic
curves). It integrates the two self-affinities, by which constant curvature
curves in similarity and equiaffine geometries are characterized in a unified
manner.
