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.

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关于一类极坐标对数美学曲线

曲线是实现复合美学曲线的基本概念，例如，组装复杂的轮廓，匹配工业设计中的特定曲率轮廓，以及在车辆机器人导航系统中构建平滑、舒适和安全的轨迹。能够对美学曲线进行编码、生成、评估和变形的新机制有望提高工业设计的吞吐量和质量。近年来，对（原木）美学曲线的研究因其在鸟蛋、蝴蝶翅膀、猎鹰飞行等自然现象以及日本刀剑和汽车等制成品中无处不在而引起了社会的关注。(log) 美学曲线呈现由直线近似的对数曲率图，和极坐标美学曲线能够在极坐标系中模拟用户定义的极切线角动力学。因此，曲率轮廓通常成为切线角的副产品。在本文中，我们扩展了极坐标美学曲线的概念，并建立了解析公式以构建具有用户定义标准的美学曲线。特别是，我们提出了满足用户定义的曲率轮廓和极切向角动力学的极坐标对数美学曲线的封闭形式解析特征。我们提供了数值例子，描绘了渲染由直线表示的对数曲率图的可行性。我们的方法可以在极坐标系中无缝表征美学曲线，