On a Class of Polar LogAesthetic Curves
Abstract
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 vehiclerobot 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 userdefined dynamics of the polar tangential angle in the polar coordinate system. As such, the curvature profile often becomes a byproduct 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 userdefined criteria. In particular, we propose the closedform analytic characterizations of polar logaesthetic curves meeting userdefined 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.
 Publication:

arXiv eprints
 Pub Date:
 July 2021
 arXiv:
 arXiv:2107.09489
 Bibcode:
 2021arXiv210709489P
 Keywords:

 Computer Science  Computational Geometry;
 Computer Science  Computational Engineering;
 Finance;
 and Science;
 Mathematics  Numerical Analysis
 EPrint:
 Accepted for presentation in (and publication in the proceedings of) The ASME 2021 International Design Engineering Technical Conferences &