Abstract Gears play a pivotal role in machine design. This paper proposes an algorithm to simplify the shapes of planar gears. This is achieved via iterative conjugation, using precise algebraic sweeps. The notion of shape simplification is introduced in a mathematically rigorous manner and it is shown that the conjugation process converges, yielding a pair of meshing gears that follow the desired motion. Simplified gear shapes may lead to improved mechanical characteristics and reduction in manufacturing costs. The generality of algebraic sweeps allows precise design of gears with freeform shapes and non-uniform motion transmission. Moreover, the computational framework proposed in this paper is versatile, with applications beyond gear design. A variety of examples from an implementation of our algorithm, that offers topological guarantees, are presented, which demonstrate the robustness and efficacy of our approach.

中文翻译：

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通过精确的代数平面扫描对齿轮设计进行共轭形状简化

摘要 齿轮在机械设计中起着举足轻重的作用。本文提出了一种简化平面齿轮形状的算法。这是通过迭代共轭实现的，使用精确的代数扫描。以数学上严格的方式引入了形状简化的概念，结果表明共轭过程收敛，产生一对遵循所需运动的啮合齿轮。简化的齿轮形状可以改善机械特性并降低制造成本。代数扫描的通用性允许精确设计具有自由形状和非均匀运动传输的齿轮。此外，本文提出的计算框架是通用的，适用于齿轮设计之外的应用。来自我们算法实现的各种示例，提供拓扑保证，