  Curves are essential concepts that enable compounded aesthetic curves, e.g.,
to assemble complex silhouettes, match a specific curvature profile in
industrial design, and construct smooth, comfortable, and safe trajectories in
vehicle-robot navigation systems. New mechanisms able to encode, generate,
evaluate, and deform aesthetic curves are expected to improve the throughput
and the quality of industrial design. In recent years, the study of (log)
aesthetic curves have attracted the community's attention due to its ubiquity
in natural phenomena such as bird eggs, butterfly wings, falcon flights, and
manufactured products such as Japanese swords and automobiles.
  A (log) aesthetic curve renders a logarithmic curvature graph approximated by
a straight line, and polar aesthetic curves enable to mode user-defined
dynamics of the polar tangential angle in the polar coordinate system. As such,
the curvature profile often becomes a by-product of the tangential angle.
  In this paper, we extend the concept of polar aesthetic curves and establish
the analytical formulations to construct aesthetic curves with user-defined
criteria. In particular, we propose the closed-form analytic characterizations
of polar log-aesthetic curves meeting user-defined criteria of curvature
profiles and dynamics of polar tangential angles. We present numerical examples
portraying the feasibility of rendering the logarithmic curvature graphs
represented by a straight line. Our approach enables the seamless
characterization of aesthetic curves in the polar coordinate system, which can
model aesthetic shapes with desirable aesthetic curvature profiles.
