On a Class of Polar Log-Aesthetic Curves

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.